Discrete diffraction for analytical approach to tightly focused electric fields with radial polarization

Optics Communications 282 (2009) 3862–3868

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Discrete diffraction for analytical approach to tightly focused electric fields with radial polarization Michele Marrocco * ENEA, via Anguillarese 301, 00123 Santa Maria di Galeria, Rome, Italy

a r t i c l e

i n f o

Article history: Received 2 April 2009 Received in revised form 19 June 2009 Accepted 25 June 2009

Keywords: Polarization Microscopy Diffraction Numerical aperture Intensity distribution

a b s t r a c t Radially polarized laser beams in high-resolution microscopy provide an effective means to reduce the focus size below the diffraction limit. Unfortunately, their theoretical manipulation is usually limited to numerical methods. Here, we demonstrate an approach that leads to analytical expressions for the focused electric fields. The approach is based on the discretization of the continuous character of diffraction taking place at the microscope focus. Comparisons with fully numerical calculation are discussed. It results that the new approach accurately reflects the distribution of light within the focal volume with relative deviations that are between 104 and 1013. Ó 2009 Elsevier B.V. All rights reserved.

1. Introduction

Eq ðq; zÞ ¼ A

Z

a2

PðhÞ cos1=2 h sin 2h J 1 ðkq sin hÞ expð2ikz sin hÞdh

a1

The use of radially polarized laser beams was introduced in optics a few years ago as one of the most ingenious ways to overcome the diffraction limit that characterizes ordinary laser microscopy [1,2]. Since then, the interest in radial polarization of electromagnetic waves has continued to grow steadily and, nowadays, applications in various fields of physics are demonstrated. Recent examples include the acceleration of massive particles [3], the design of a plasmonic lens [4], measurements with THz radiation [5], tip-enhanced Raman microscopy [6], second harmonic generation microscopy [7] and the list could continue with many other examples that are available in the specialized literature. In high-resolution microscopy, radially polarized laser beams are specifically employed with the purpose of reducing the spot size near focus. This is in line with the original proposal [1,2] and studies concerning the optical conditions leading to spot sizes well below the diffraction limit have been reported [8,9]. These works are based on the diffraction theory developed by Youngworth and Brown [10] (which is, in turn, derived from the seminal work of Richards and Wolf [11]). Following this theory, the vectorial structure of the focused electric field E(q,z) of radially polarized beams resembles what is seen for linear polarization [11]. In particular, considering an annular pupil with angular apertures between a1 and a2, the field with wave vector k can be reconstructed through the projections onto the radial plane (x, y) and the axial direction z according to * Tel.: +39 06 3048 3345; fax: +39 06 3048 4811. E-mail address: [email protected] 0030-4018/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2009.06.055

Ez ðq; zÞ ¼ 2iA

Z

ð1Þ a2

a1

2

PðhÞ cos1=2 h sin h J 0 ðkq sin hÞ expð2ikz sin hÞdh ð2Þ

where k is the wave vector amplitude, q = (x2 + y2)1/2 is the radial coordinate, A is the field amplitude and P(h) is the pupil function (otherwise known as apodization function) that depends on the variable angular aperture h. The physical meaning of Eqs. (1) and (2) is clear: the field components Eq and Ez are the resultant of the interplay between the propagation function (i.e., the plane wave with the phase term) and the corresponding radial function (i.e., the Bessel function) that describes the spreading of the light within planes lying orthogonal to the optical axis (or axial direction coincident with the z axis). The product of these two functions is modulated by the spatial structure P(h) of the light beam at the pupil, plus the residual effect of the focusing lens mediated by the sine and cosine functions. It is finally understood that the vectorial structure of the field is reconstructed ^ q þ Ez ðq; zÞv ^ z with v ^q through the relationship Eðq; zÞ ¼ Eq ðq; zÞv ^ z unit vectors belonging, respectively, to the radial plane and and v the axial direction. It is an immediate conclusion that Eqs. (1) and (2) can only be treated numerically. This means that precise reconstruction of the focal region is subjected to numerous calculations needed to cover the significant spatial ranges of the two coordinates q and z. In other words, an accurate description of the focused field is

M. Marrocco / Optics Communications 282 (2009) 3862–3868

possible by means of a two-dimensional grid where the spatial nodes are sufficiently close to each other and each node is associated with numerical solutions of Eqs. (1) and (2). In an attempt to avoid the difficulty of purely numerical approaches, research is being undertaken to look for analytical expressions of the fields. For instance, the Lax series (i.e., the solution of the wave equation written as polynomial expansion with respect to the diffraction angle) is used by some authors [12–14]. Others suggest an analytical approach based on the vector angular spectrum of the focused beam [15]. Here, we propose an alternative method that aims at the discretization of the continuous character of diffraction. The strategy behind this attempt is very simple and can be summarized as follows. The phenomenon of diffraction is optically conceptualized as a combination of plane waves undergoing propagation along the z axis and simultaneous spreading in radial planes. This is manifest in the common diffraction theory of linearly polarized beams [11] and, looking at Eqs. (1) and (2), one can conclude that radially polarized beams make no exception [10]. The simplicity of this pictorial view contrasts with the complexity created by the continuum of wave vector projections on the longitudinal plane (note that the role of k sin h in the integrals of Eqs. (1) and (2) highlights the importance of the k vector projected onto the longitudinal or radial plane). But, we point out that the continuity of h is not so important. As a matter of fact, it is possible to establish a golden rule that would enable us to discard those angles that do not contribute significantly to the whole process of diffraction. In doing so, we hope that a selection of a limited number of angles (i.e., a limited number of wave vector projections) will be enough to describe diffraction at the focus of radially polarized beams. How to achieve this goal will be shown in the next Section. 2. Method The description of the method starts with the observation that the integrals in Eqs. (1) and (2) can be simplified by means of Taylor expansions of the radial and propagation functions. Since we are interested in what happens around the focal region, then we can transform the Bessel function according to 1 X ð1Þ ðkq sin hÞ l

J m ðkq sin hÞ ¼

l¼0

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examine the most basic shape of spatial profiles. Thus, the pupil function is

PðhÞ ¼ exp½ðb sin h= sin a2 Þ2 

ð7Þ

where b is the so-called filling factor (i.e., the ratio between the aperture radius and the beam waist at the aperture). Unfortunately, the Gaussian character of Eq. (7) impedes the search for an analytical solution of the integrals in Eqs. (5) and (6). As a result, we are forced to circumvent this problem. Let us change the variable of integration. We call f = sin2h and the fields become

Eq ðq; zÞ ¼ A

1 1 X ð1Þl ðkqÞ2lþ1 X ð2ikzÞt cq ð2l þ tÞ 2lþ1 t! l!ðl þ 1Þ! t¼0 l¼0 2

ð8Þ

Ez ðq; zÞ ¼ iA

1 1 X ð1Þl ðkqÞ2l X ð2ikzÞt cz ð2l þ tÞ 2l 2 t! 2 l! t¼0 l¼0

ð9Þ

where the coefficients cq and cz are defined according to

cq ðqÞ ¼ cz ðqÞ ¼

Z

sin2 a2

exp½ðb= sin a2 Þ2 fð1  fÞ1=4 fðqþ1Þ=2 df

ð10Þ

exp½ðb= sin a2 Þ2 fð1  fÞ1=4 fðqþ1Þ=2 df

ð11Þ

sin2 a1

Z

sin2 a2

sin2 a1

with q = 2l + t. It is worth noticing that the coefficients cq and cz depend on the optical parameters characterizing the experimental setup. In other terms, the coefficients are calculated once and for all when a1, a2 and b are given. This is the only numerical calculation that is needed in this procedure and the values of cq(q) and cz(q) can be tabulated and stored in view of the analytical elaboration of the fields. The ultimate solutions of Eqs. (8) and (9) would be very easy if the series in each field component were separable and convergent. To this end, we suggest a suitable approximation to the coefficients cq(q) and cz(q). We suppose that Eqs. (10) and (11) can be regarded as linear combinations of powers of the sine of certain angles aq,h and az,h

cq ðqÞ ¼

Nq X

aq;h ðsin aq;h Þq

ð12Þ

az;h ðsin az;h Þq

ð13Þ

h¼1 2lþm

22lþm l!ðm þ lÞ!

ð3Þ

cz ðqÞ ¼

Nz X h¼1

whereas the phase term becomes 1 X expð2ikz sin hÞ ¼ ð2ikz sin hÞt =t!

ð4Þ

t¼0

In this manner, the field components take the following forms 1 1 X ð1Þl ðkqÞ2lþ1 X ð2ikzÞt 2lþ1 t! l!ðl þ 1Þ! t¼0 l¼0 2 Z a2 PðhÞ cos3=2 hðsin hÞ2lþtþ2 dh 

Eq ðq; zÞ ¼ 2A

ð5Þ

a1 1 1 X ð1Þl ðkqÞ2l X ð2ikzÞt Ez ðq; zÞ ¼ 2iA 2l 2 t! 2 l! t¼0 l¼0 Z a2 PðhÞ cos1=2 hðsin hÞ2lþtþ2 dh 

where Nq and Nz indicate the total numbers of terms necessary to approximate Eqs. (10) and (11) with Eqs. (12) and (13) within a certain accuracy and aq,h, az,h are the coefficients of the two linear combinations. If Eqs. (12) and (13) held true, then the fields would have simple solutions with a direct connection to the original functional structure of Eq and Ez. Indeed, the fields would be represented by linear combinations of plane waves with Bessel functions governing the distribution of light in radial planes, that is

Eq ðq; zÞ ¼ A

Nq X

aq;h J 1 ðkq sin aq;h Þ expð2ikz sin aq;h Þ

ð14Þ

az;h J 0 ðkq sin az;h Þ expð2ikz sin az;h Þ

ð15Þ

h¼1

Ez ðq; zÞ ¼ iA ð6Þ

a1

The passage from Eqs. (1) and (2) to (5) and (6) is guaranteed by the theorem relative to the integration of uniformly convergent series (theorem 11.4 in [16]) and, therefore, we can integrate the series appearing in Eq and Ez term by term. We assume the case of Gaussian beams that are recurrent in microscopy. In principle, other types of beams can be treated in the following procedure, but it seems a compulsory choice to

Nz X h¼1

The meaning of Eqs. (14) and (15) is manifest. With reference to the initial expression in Eqs. (1) and (2), we have operated a selection of angles (or wave vector projections on the radial plane). As a consequence, the continuity of sin h has been broken down into discrete contributions of sin aq,h and sin az,h. Since the amplitudes aq,h, az,h and the angles aq,h, az,h are known initially (they are determined by the parameters a1, a2 and b that are established by the experimental details), the spatial dependences in Eqs. (14) and (15) are

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analytical and can be used in the treatment of optical problems involving tight focusing of radially polarized beams. It is important to repeat that the essential premise of our reasoning is that Eqs. (12) and (13) are obeyed. The method revolves around this crucial approximation and a first hint of validity is suggested by the application of the mean value theorem for integration (Theorem 3.15 in [16]) to cq(q) and cz(q) in Eqs. (10) and (11). This theorem suggests that the power dependence on the q parameter in the integrand of Eqs. (10) and (11) dominates the result of the integration and, as a consequence, the approximation in Eqs. (12) and (13) seems well founded. However, to verify in a rigorous manner the whole scheme presented here, we can now illustrate how a practical example can be handled. 3. Results and discussion Having defined the rules of the transition from continuous to discrete representation of diffraction of a tightly focused Gaussian

Fig. 1. Logarithmic plot of cq(q) and cz(q) according to Eqs. (10) and (11) calculated for an annular aperture with maximum NAmax = 1.4 and minimum NAmin = 1.0. The filling factor b is assumed unitary. The vertical axis is rescaled with a factor of 104.

beam with radial polarization, we next show how to apply these rules to the specific example of a lens with an annular aperture (maximum NAmax = 1.4, minimum NAmin = 1.0) associated with a filling factor of b = 1. We additionally suppose unitary amplitude A and index of refraction equal to 1.5. Such a choice of the numerical values of these optical parameters is dictated by the idea that it is interesting to consider the case of high-resolution microscopy with an annular aperture that guarantees an important breaking of the diffraction limit [8]. On the other hand, another choice would not alter the general application of the method developed before (except for the different shapes of the focus). With the chosen parameters, the coefficients in Eqs. (10) and (11) are readily calculated. They are shown in the logarithmic plot of Fig. 1. The quasi-linear decays support the idea that they can be effectively approximated by a set of power law dependences and this corresponds to the interpretation of cq(q) and cz(q) as linear combinations of powers evoked in Eqs. (12) and (13). The data of Fig. 1 are thus used to determine the parameters Nq, Nz, aq,h, az,h and the angles aq,h, az,h. In particular, we determine Nq and Nz according to the criterion that, away from the focal zone (that is, at fixed values of q and z far from the exact focus), the relative deviation between numerical and analytical calculation of the intensity is negligible (about 104, in this paper). Following this specific criterion and taking kz = ± 10 and kq = 10, we find that the number of projections is surprisingly small (Nq = Nz = N = 12) in comparison with the complex structure of a focus of a high NA microscope. The corresponding structure of the electric fields is reproduced in Fig. 2. Here, the real and imaginary parts of the two components Eq and Ez of the total electric field E are shown as functions of the dimensionless variables kq and kz. In these plots, beyond the obvious oscillating nature of the field amplitude, the higher values of the longitudinal field Ez with respect to the radial field Eq underline what is customarily considered as one of the prominent characteristics of radially polarized beams [2,8,10]. But the result of Fig. 2 has yet to be compared with the ordinary numerical calculation of Eqs. (1) and (2). The comparison is given in Fig. 3, where the differences Re[(Eq)Num  (Eq)Anal], Im[(Eq)Num  (Eq)Anal], Re[(Ez)Num  (Ez)Anal]

Fig. 2. Three-dimensional plots of the real and imaginary parts of the fields according to Eqs. (14) and (15) calculated for the data of Fig. 1 and Nq = Nz = N = 12. In the upper plots, Re[Eq] and Im[Eq] are shown in dependence on the dimensionless variables kq and kz. Lower plots, same dependences for Re[Ez] and Im[Ez].

M. Marrocco / Optics Communications 282 (2009) 3862–3868

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Fig. 3. Differences between analytical and numerical calculations of the electric fields. The analytical fields (Eq)Anal and (Ez)Anal are based on Eqs. (14) and (15) calculated for the data of Fig. 1, whereas numerical solutions (Eq)Num and (Ez)Num are obtained by means of Eqs. (1) and (2). Plots in the panels (a) and (b) represent, respectively Re[(Eq)Num  (Eq)Anal] and Im[(Eq)Num  (Eq)Anal] as functions of kq and kz. Lower plots in (b) and (c) contain, instead, the behavior of Re[(Ez)Num  (Ez)Anal] and Im[(Ez)Num  (Ez)Anal].

and Im[(Ez)Num  (Ez)Anal] between numerical and analytical calculations are illustrated. As apparent, the relative deviations are very small. For instance, considering the radial field Eq at the border of the plotted region (where the maximum difference is located), the relative deviation is found to be around 103 for the real part and 4104 for the imaginary part. More importantly, the relative deviations are orders of magnitude smaller in the proximity of the field maximum (e.g., 71012 at kq ffi 2 and kz ffi 0). Similar conclusions can also be drawn for the longitudinal field Ez. In this case, the relative deviations for the real and imaginary parts are, respectively around 108 and 106 at the edges of the plotted region (kq ffi 10 and kz ffi 10). Near the field maximum, the result shows that the relative deviation goes down to a staggering 1013. The good agreement between analytical and numerical calculations of the fields is suggestive of an analogous agreement on the calculation of field intensities. These are reported in Fig. 4. In the upper plot, the intensity jEqj2 shows another characteristic of radially polarized beams, namely the vanishing values of the radial intensity along the axial direction. On the contrary, the longitudinal intensity jEzj2 in the lower plot has its maximum along the axial direction with values that are about four times larger than the maximum values of jEqj2. The data of Fig. 4 can be further elaborated to compose the total intensity, which is given in the upper plot of Fig. 5. The lower plot quantifies, instead, the difference D between analytical and numerical calculations. It turns out that the relative deviation is generally very small. In particular, we find values around 104 (absolute deviation D is about 107) at the edge of the plotted region (kq ffi 10 and kz ffi ± 10), but the relative deviation decreases to a value of about 1013 around the origin (kq ffi 0 and kz ffi 0) where the exact focus lies. Promising and favorable as these comparisons may seem, we should also discuss the improvement in the calculation speed, which depends on the number of points used to section the twodimensional plane (kq, kz). About 2000 points were chosen to map the fields and the associated intensities in the results shown

earlier. As expected, numerical treatment based on Eqs. (1) and (2) was always slower. For instance, the computing time of the total intensity, calculated according to the standard procedure based on numerical methods, amounted to about 42 s of the CPU time of

Fig. 4. Behavior of jEqj2 and jEzj2 according to Eqs. (14) and (15) calculated for the data of Fig. 1.

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Fig. 5. Upper plot: distribution of light intensity calculated for the total analytical electric field EAnal(q,z) containing the two components (Eq)Anal and (Ez)Anal shown before. Lower plot: deviation D = jENum(q, z)j2  jEAnal(q,z)j2 between the distributions of light intensity obtained from the standard and analytical derivations.

a common PC (2.40 GHz, 2 GB RAM) equipped with Mathematica 6.0 (Wolfram Research, Inc.). By contrast, the proposed analytical method took about 5 s to complete the analogous calculation shown in the upper part of Fig. 5. Although the application of numerical methods is still reasonable (few tens of seconds are manageable, of course) in linear microscopy, the advantage of analytical approaches [12–15, and this work] is remarkable for future uses of radially polarized beams in multi-photon microscopy [17]. In this latter context, the non-linear character of the interaction between fields and matter is of difficult theoretical treatment if any attempt is made by following the standard method. More precisely, the recurrent use of numerical integration in non-linear problems poses a serious trouble in the description of the coupling of the electric fields with the non-linear response of the optical medium (an example in this regard for a third-order non-linearity is in [18]). For this reason, the analyticity of the focused fields is expected to improve the theoretical handling and modeling of nonlinear microscopy performed with tightly focused radially polarized laser beams and, depending on the order of the non-linearity, the computing time should be hundreds of times shorter. 4. Comparison with other methods Having explored the agreement between the numerical treatment of Eqs. (1) and (2) and their analytical reduction shown in Eqs. (14) and (15), it becomes instructive to contrast the main result of this paper with similar findings reported in the literature concerning analytical representation of radially polarized light beams. In the past years, considerable attention has been given to the approach based on the Lax series [12–14]. In particular, the series is useful to calculate the expansion of the radial and axial electric fields with respect to the diffraction angle e (defined as the ratio between the Gaussian beam waist w0 and the Rayleigh length 2 zR ¼ kw0 =2) [12,13]. Analogously, in the approach that transforms the series into a complex-source-point spherical wave [15], the expression of the electric fields is again depending on e. A first conclusion is that the limit of validity of the two solutions to the prob-

Fig. 6. Behavior of jEqj2 and jEzj2 according to the method based on the Lax series explained in Refs. [19,20]. The plots refer to the exact focus at z = 0. Besides the expected qualitative behavior near the optical axis (qnorm = 0), remarkable oscillations are found for qnorm > 1.

lem is given by the inescapable condition on the convergence of the series, that is e < 1 or w0 > k/p. Secondly, the analytical treatments suggested by these authors do not contemplate the role of an annulus that is so important to overcome the diffraction limit [1,2,8,9] and, putting aside the seemingly superiority of the traditional Lax series in comparison with the complex-source-point spherical wave method [19], we can prove that the Lax series fails in the detailed reconstruction of the focus obtained from the standard diffraction theory summarized in Eqs. (1) and (2) [10]. For instance, we take the clear description by Salamin [20], who provides the details of the method delineated in previous works [12,13]. The first fifteen terms of the Lax series are deemed sufficient to reproduce the electric fields of a radially polarized Gaussian beam focused by means of a lens of high NA (but without annular aperture!!!). Considering the value e = 0.8 of the diffraction angle, we find the plots of Fig. 6, which are relative to the focused intensities of the radial and axial fields plotted in correspondence of the exact focus (z = 0) as functions of qnorm = q/w0. These results are a mere reproduction of what is reported by Salamin in Fig. 3 of Ref. [19]. The general feature of a vanishing radial component along the optical axis (Fig. 6(a) near qnorm = 0) is found jointly with the maximum intensity of the longitudinal component (Fig. 6(b) near qnorm = 0). Nonetheless, the plots are troubled by multiple oscillations between qnorm = 1 and qnorm = 4. These concentrated oscillations remain somewhat of a mystery because standard diffraction theory for the focusing of the fundamental Gaussian beam (that is, without annular aperture) does not predict them [10,11,21]. To verify this point, the numerical solutions of Eqs. (1) and (2) are reported as continuous lines in Fig. 7 for a NA value corresponding to the diffraction angle of Fig. 6 (note that NA = ne, with n the index of refraction [21]) and for a filling factor b = 3 that guarantees the clear Gaussian profile of the beam. Although the above-mentioned partitioning of light between radial and axial components near the optical axis (kq = 0) is replicated here, the remarkable oscillations of Fig. 6 do not occur, as expected

M. Marrocco / Optics Communications 282 (2009) 3862–3868

Fig. 7. Behavior of jEqj2 and jEzj2 according to the standard diffraction theory (line) for the NA value equivalent to the diffraction angle of Fig. 6. The squares reproduce the results based on the analytical approach of Eqs. (14) and (15). As expected from the work of Youngworth and Brown [10], the oscillations of Fig. 6 are absent.

from the original work of Youngworth and Brown [10]. More importantly, the analytical treatment based on Eqs. (14) and (15) captures correctly the behavior of diffraction theory (see the points in Fig. 7).

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A further comparison can be established with the work of Sherif and Török [22], who succeeded in expanding the focused electric fields of a radially polarized Gaussian beam in terms of circulate prolate spheroidal functions. The method is rather tortuous but, for the sake of completeness, we have to consider it. The main result lies in Eqs. 26(a) and (b) of Ref. [22]. These two equations illustrate the field distribution Eq and Ez of the radial and axial components and, as apparent, the calculation of Eqs. (1) and (2) is transformed into the product of two series. One takes into account the infinite group of Bessel functions of the first kind and the other results from the infinite set of prolate spheroidal functions multiplying some coefficients whose values depend on additional mathematical work. This includes the use of Chebyshev polynomials, the determination of eigenvalues of the finite Hankel transform operator and so on. From this short summary, it turns out that the method is not of first choice. To begin with, the calculation of two infinite series is not faster and simpler than the numerical approach to the solution of the one-dimensional integrals of Eqs. (1) and (2). In the second place, the mathematical structure is very arduous and defies easy application to real problems. For this reason, the authors had to introduce some simplifications. For instance, in their numerical example, they squeeze the whole calculation of the two infinite series into 29  9 = 261 terms only (see Eqs. (30a) and (30b) of Ref. [22]). The first 29 terms come from the truncated series for the Bessel functions. The remaining 9 terms are extracted from the series for the prolate spheroidal functions. Under this simplification, the final acquisition is that the percentage error between the standard numerical solution of diffraction theory and the proposed method for a lens of NA = 0.966 is on the order of 102 (see the plots for the percentage error in Figs. 14 and 16 of Ref. [22]). For the same optical conditions, the result of the method based on discrete diffraction (i.e., Eqs. (14) and (15) of this work) is shown in Fig. 8, where the density plots of jEqj2 and jEzj2 in the focal plane (z = 0) are displayed. Notably, for the assumed NA, the maximum value max{jEqj2} of

Fig. 8. Behavior of jEqj2 and jEzj2 in the focal plane (z = 0) of a lens with NA = 0.966. The electric fields are calculated by means of Eqs. (14) and (15) for an under-filled lens (b > 1) so that the main portion of the Gaussian beam is not stopped by the lens aperture. The profiles of jEqj2 and jEzj2 at y = 0 are reported on the right-hand side of Fig. 8. The units of the vertical scales of the two profiles are arbitrary but comparable.

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M. Marrocco / Optics Communications 282 (2009) 3862–3868

(15). By contrast, as stated earlier, 261 terms are assumed in the applicative example of Ref. [22]. 5. Conclusions

Fig. 9. Percentage error defined as the percentage deviation between the analytical and numerical calculation of jEqj2 + jEzj2 (see inset). The analytical result is based on Eqs. (14) and (15), whereas the numerical result stems from Eqs. (1) and (2). Both results refer to the optical conditions of Fig. 8. The horizontal axis is in units of wavelength and direct comparison with the percentage error of Fig. 14 in the work by Sherif and Török [22] can be established.

the radial intensity is found to be about two times the maximum value max{jEzj2} of the axial intensity (compare the profiles of Fig. 8). This is in complete agreement with the independent work by Yew and Sheppard [8], where the dependence of the ratio max{jEqj2}/max{jEqj2} on NA is provided for various types of illumination (Fig. 2 in Ref. [8]). There, one can notice that the ratio max{jEqj2}/max{jEqj2} resides around 0.5 for a radially polarized Gaussian beam focused with a lens of 0.9 6 NA 6 1. Based on this further evidence, the results of Fig. 8 are finally composed in Fig. 9 to determine the percentage error of the total intensity in the focal plane, so that a quantitative comparison between this work and Ref. [22] is established. In the horizontal axis, the radial coordinate is evaluated relatively to the wavelength and, in this manner, Fig. 9 can be directly contrasted with the percentage error of Fig. 14 in Ref. [22]. It can be clearly seen that the method of discrete diffraction is quantified by means of a percentage error of several orders of magnitude smaller than the method conceived by Sherif and Török. In effect, Fig. 9 demonstrates that the percentage error is practically zero up to q  k. After this limit, the error increases but is still rather low (on the order of 104) in comparison to the error calculated for the method of prolate spheroidal functions, which yields percentage errors of about 102 across the entire horizontal axis. More convincingly, the much better performances of the method based on discrete diffraction are built around 12 terms of the summations appearing in Eqs. (14) and

Analytical treatment of radially polarized beams useful in highresolution microscopy has been demonstrated. Based on the simple conceptual understanding of tight focusing depicted in the work of Youngworth and Brown [10], the analyticity stems from the suppression of the continuous character of diffraction. In this way, only a set of diffraction angles (or wave vector projections) is made to survive and becomes optically relevant to the description of the focused fields. The criterion leading to the simplification is described in detail and direct comparisons with the standard treatment corroborate the argument developed in this work. In conclusion, the current proposal can be viewed as an alternative method of dealing with analytical focused electric fields with radial polarization, a subject that includes already a significant number of works [12–15] in spite of its very recent introduction. We hope, additionally, that the advantage of having fast computing times will soon be put to good use in the field where analytical expressions for the fields appear extremely useful (i.e., non-linear microscopy). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

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