Monochromatic electromagnetic fields with maximum focal energy density

Moore et al. Vol. 4, No. 10 /October 007 /J. Opt. Soc. Am. A 3115 Monochromatic electromagnetic fields with maximum focal energy density Nicole J. Moore, 1, * Miguel A. Alonso, 1 and Colin J. R. Sheppard,3 1 The Institute of Optics, University of Rochester, Rochester, New York 1467, USA Division of Bioengineering, National University of Singapore, 9 Engineering Drive 1, Singapore 117576 3 Department of Diagnostic Radiology, National University of Singapore, 5 Lower Kent Ridge Road, Singapore 119074 *Corresponding author: ncarlson@optics.rochester.edu Received April 10, 007; accepted June 5, 007; posted June 13, 007 (Doc. ID 81961); published September 10, 007 The monochromatic electromagnetic fields that achieve maximum focal energy density for a given input power and directional spread are found through a variational approach. It is found that the polarization of the fields at the focal point must be perpendicular to the main direction of propagation. Parametric expressions relating the directional spread and focal energy density of these fields (and others that achieve stationary focal irradiance) are found, both in the optical (electric field only) and in the electromagnetic cases. 007 Optical Society of America OCIS codes: 60.060, 60.110, 350.740. 1. INTRODUCTION In a previous paper [1] we found the fundamental upper bound on the focal irradiance of a monochromatic scalar field given its input power and directional spread, as defined by Alonso and Forbes []. The results are now extended to monochromatic electromagnetic fields, i.e., vector fields. For many applications, such as confocal and differential interference contrast microscopy, the polarization properties of focal fields are of great importance. As such, this extension provides new insight into the fields that maximize the focal irradiance for a given directional spread. The variational techniques employed in [1] were used to jointly minimize a focal-irradiance-based measure of spatial spread and the previously mentioned measure of directional spread. The focal-irradiance-based measure of spatial spread was chosen to characterize the ratio of the focal irradiance to the input power because of its simple geometric interpretation in the paraxial limit: the radius of a cylinder with volume equal to the input power and height equal to the focal irradiance. The measure of directional spread has a simple geometric interpretation in the direction cosine space in terms of the centroid of the Ewald sphere weighted by the radiant intensity. Additionally, in the paraxial limit, this measure becomes equal to the standard deviation in the far field. Detailed discussions of this measure can be found in [1 3]. Both the focal-irradiance-based measure of spatial spread and the measure of directional spread are herein generalized for vector fields such that they may be jointly minimized as well. Unlike in [1], only three-dimensional fields are considered here. The resultant bounds depend upon whether the effects of only the electric field (valid in the optical regime) or the fully electromagnetic field are considered. In the case in which only the electric field is considered, an upper bound is obtained for the irradiance (electric energy density) at a point (which is identified herein as the focus) of a monochromatic electric field for a given input power and directional spread. For the electromagnetic case, an upper bound on the total focal energy density given the input power and directional spread is found.. PRELIMINARIES A free monochromatic electric field can be expanded in terms of plane waves as Er =S 3 Auexpiku rd, where u is a unit vector indicating the direction of propagation of the plane waves and the integral is over the complete unit sphere S 3 of directions of u. Note that this field does not contain evanescent components. Here Au gives the amplitude of the plane waves that compose the field. The transversality of the plane waves restricts Au to be perpendicular to its argument, i.e., u Au=0. The corresponding monochromatic magnetic field can be written similarly as cbr =S 3 u Auexpiku rd, where c is the speed of light. Since only monochromatic fields are considered, the wavenumber k is set to unity henceforth so that all distances are in units of reduced wavelength. The frequency is also set to unity. In so doing, the need to carry the constant c throughout the calculations is eliminated. Additionally, all integrals are taken to be over the unit sphere S 3. 1 1084-759/07/103115-8/$15.00 007 Optical Society of America

3116 J. Opt. Soc. Am. A / Vol. 4, No. 10 / October 007 Moore et al. 3. OPTICAL FIELDS WITH MAXIMUM FOCAL IRRADIANCE In the optical regime, detectors are not significantly affected by the presence of the magnetic field, so it is sufficient to consider the electric field in the definitions that follow. In this case, it is easy to generalize the spread measures used in [1] from scalar to vector fields. The measure of directional spread is changed only by simply replacing Au with Au =A * u Au, i.e., = arccosu, where u is the centroid of the unit sphere weighted by the radiant intensity, Au : u = 1 E uau d. Here the total power in the electric field is given by E = Au d. 3 4 5 P u A * = cos cos E sin Au, 9 where the fact that P u Au=Au was used to simplify the expression. The functional derivative of the spatial spread is simply E D R P u A * E D = R E 3 D E Au R 8E P ue0. 10 Equations (9) and (10) can be substituted into Eq. (8) to solve for the plane-wave amplitude, which is given by an expression of the form for nonnegative a, where Au = coth a cos P ue0, D E R 3 = sin, 8 11 1a The direction of u corresponds to the beam s main propagation direction, defined to be along the positive z axis so that the magnitude of Eq. (4) reduces to coth a = D R E sin + cos. 1b u = u z = 1 E cos Au d, where is the angle between u and the z axis. Additionally, the origin is chosen to coincide with the focus. The focal-irradiance-based measure of spatial spread is equally simple to generalize from the scalar case, yielding A * u Aud D E R =4 E E0 =4, A * ud Aud where, in the second step, E0 is calculated by evaluating Eq. (1) at the origin. As in [1], a variational procedure is used to search for the fields that simultaneously minimize D E R and, i.e., that maximize the focal irradiance for a given directional spread. The basis of this approach is to constrain a positive linear combination of D E R and to be stationary under infinitesimal changes in the plane-wave amplitude: P u E A * + D R A * = 0, 8 where 0 is the Lagrange multiplier. It is important to remember that the components of the plane-wave amplitude may not vary independently, since Au must remain perpendicular to u. Therefore, only the transverse part of the functional derivative need be considered. This is indicated by the inclusion of the projection matrix onto the plane perpendicular to u, P u, which operates on a vector v as P u v=uvu=v uu v. The first term of Eq. (8) yields 6 7 The electric field at the focus, E0, can be written as a magnitude E 0 times a (possibly complex) unit focal polarization vector ê (i.e., ê * ê=1), so the expression for the plane-wave amplitude can be written as Au = coth a cos E 0P u ê = coth a cos E 0u ê u. 13 When Eq. (13) is integrated over all directions, it gives the electric field at the focus, which should be equal to E 0 ê, in order to be self-consistent. After performing the integration, the electric field at the focus is found to be where E0 = E 0 1 e + e z ẑ, 1 =acoth a +1 coth a, =4coth a a csch a, 14 15a 15b and e =e x xˆ +e y ŷ. It is easy to see that Eq. (15) can be selfconsistent only when ê is either exclusively transverse (i.e., perpendicular to ẑ) or exclusively longitudinal (i.e., parallel to ẑ), since 1 except in the limit a 0. Therefore, the coefficient 1 must equal unity in the case of a transverse focal electric field, while the coefficient must equal unity in the case of a longitudinal focal electric field. These constraints serve to find the normalization factor in each case [4]. A. Transverse Focal Electric Field We first consider the case for which the direction of the focal electric field lies entirely in the x y plane, such that ê=e. Calculation of the total energy for these fields is straightforward, yielding

Moore et al. Vol. 4, No. 10/October 007/J. Opt. Soc. Am. A 3117 E =4 cosh a a coth ae 0, 16 D R E = cosh a a coth a acoth a +1 coth a. 17 upon consideration of the fact that e * e =1 in this case. Substituting for from Eq. (15a) with 1 =1 into Eq. (16) and noting that E0 =E 0, the spatial spread is found to be Similarly, the angular spread, calculated from Eq. (3), is given by = arccos 3 coth acosh a sinh a a cosh a a coth a cosh a sinh a + a cosh a a coth a. 18 It is easy to see by inspection that the upper bound of the focal irradiance is not simply proportional to the directional spread but must be expressed parametrically. This is not surprising, as this was the case for threedimensional scalar Helmholtz fields as well [1]. In the paraxial limit a, D R E 1 a expa, a exp a, 19 which correspond to the values obtained upon taking the same limit in the three-dimensional scalar case, as expected. In the nonparaxial limit a 0, this field corresponds to an electric dipole (or combination of electric dipoles) placed in any transverse direction at the origin. The irradiance distributions for such a field are rotationally symmetric if the focal field is chosen to have circular polarization, i.e., if ê is given, for example, by xˆ +iŷ/. This case is plotted in Fig. 1 over the y=0 plane for coth a=1.01, 1.1, and. These irradiance distributions look similar to the irradiance distributions found in the scalar case, especially when the fields approach the paraxial limit. B. Longitudinal Focal Electric Field Consider now longitudinal focal electric fields, for which ê=ẑ. Given their rotational symmetry, these fields correspond to fields with radial electric polarization. In this case, the total energy is given by E =8 a coth a 1E 0. 0 Substituting for from Eq. (15b) with =1 in Eq. (0), the spatial spread given by Eq. (10) is found to be D R E = a coth a 1 coth a a csch a. 1 The directional spread is given by = arccos a3 coth a 1 3 coth a a coth a 1. Fig. 1. (Color online) Irradiance Er of an optimal electric field with transverse circular polarization at the focus for a given total power E for (a) coth a=1.01, (b) coth a=1.1, and (c) coth a= in the y=0 plane. Note that the focal irradiance increases as coth a (and consequently the directional spread of the field) increases. As with the transverse case, the upper bound of the focal irradiance must be expressed parametrically. Figure 4 in Section 5 shows that the transverse electric case has better joint spatial directional localization than the longitudinal case. In other words, for the same power and directional spread, the transverse case allows greater focal irradiance than the longitudinal case; therefore, the transverse case sets the fundamental bounds that must be satisfied by any field. The interest in radially polarized

3118 J. Opt. Soc. Am. A / Vol. 4, No. 10 /October 007 Moore et al. B = u Au d = Au d = E. 5 However, the spatial spread measure must now be based on the total electromagnetic energy density at the focus, E0 +B0, instead of only the electric energy density (the irradiance): Fig.. (Color online) Irradiance Er of an optimal electric field with longitudinal polarization at the focus for a given total power E for coth a=1+110 13, which corresponds to =0.633, which is the same directional spread used in Fig. 1(a). The irradiance distributions for varying values of coth a are similar in appearance, although the focal irradiance increases with increasing coth a but always remains less than that of an optimal electric field with transverse polarization and the same directional spread. fields is sparked by the fact that although such fields have larger focal spots than equivalent linearly polarized fields, the z component of these fields has been found to create tighter focal spots [5 7]. Here, however, all components of the field have been included. In the paraxial limit, when a1, the measures of spread in Eqs. (1) and () for these radially polarized fields can be found to be approximately given by D E R a, 1 a, 3 so that D R E =. 4 In other words, the focal irradiance of a paraxial radially polarized field has an upper bound proportional to the total power and the square of the directional spread. Alternatively, knowledge of the focal irradiance of a radially polarized field implies an easily calculable minimum directional spread in the paraxial limit. The irradiance distribution over the y=0 plane for a longitudinal focal field with maximum focal irradiance is plotted in Fig. for coth a=1+110 13, which corresponds to directional spread =0.633. Note that the focal irradiance is less than that of a transverse field with the same, corresponding to coth a=1.01 [see Fig. 1(a)]. 4. ELECTROMAGNETIC FIELDS WITH MAXIMUM FOCAL ENERGY DENSITY Let us now consider the case where the effects of the magnetic field are taken into account. It is only slightly more complicated to generalize the spread measures from [1] to this case than to the previously discussed optical case. It is easy to see that the total power in the magnetic field equals that of the electric field: D EM R =4 E + B E0 + B0 =8 E E0 + B0. The functional derivative of this measure is given by EM EM D R D R D EM P u A * = E Au R 3 16E P ue0 + B0 u. 6 7 The directional spread measure, on the other hand, does not change: = arccos Au cos d + u Au cos d = arccos Au cos d E E + B, 8 and as a result, neither does its functional derivative, which is given by Eq. (9). It is again convenient to define the focal electric and magnetic fields in terms of magnitude and complex direction: E0 = E 0 ê, B0 = B 0 bˆ, 9a 9b where ê and bˆ are complex unit vectors such that ê * ê =bˆ * bˆ =1. Equations (9) and (7) (9) can be substituted into the equation P u EM A * + D R A * = 0 30 in order to solve for the plane-wave amplitude, yielding Au = coth a cos E 0u ê u + B 0 bˆ u, where, in this case, coth a = D EM R 3 = sin, 16 D R EM sin + cos. 31 3a 3b In order to maintain self-consistency, the values for the focal electric and magnetic fields, calculated by Eqs. (1)

Moore et al. Vol. 4, No. 10 /October 007 /J. Opt. Soc. Am. A 3119 and (), respectively, must equal those defined in Eqs. (9). After substitution of Eq. (31), Eqs. (1) and (), when evaluated at the origin, give E = B =8 coth a a csch a, 40 coth a E0 = E 0 ê = E 0 1 e + E 0 e z ẑ + B 0 3 bˆ ẑ, B0 = B 0 bˆ = B 0 1 b + B 0 b z ẑ E 0 3 ê ẑ, 33a 33b and the resultant spatial spread is given by D R EM = 8coth a a csch a respectively, where 1 and are the same as in Eqs. (15) and 3 =4a coth a 1. By using b =bˆ b z ẑ, Eq. (33b) can be rewritten as bˆ = 1 1 1 b z ẑ E 0 B 0 3 1 1 ê ẑ. 34 35 Upon substituting this expression into Eq. (33a), the magnitudes of the focal fields drop out, and the following selfconsistency expression is found for the direction of the focal electric field: 3 ê = 1 + + e z ẑ. 1 1e In the same fashion, it can be shown that 3 bˆ = 1 + + b z ẑ. 1 1b 36a 36b It is evident from Eqs. (36) that the electric and magnetic focal fields must be either simultaneously transverse or simultaneously longitudinal, since the coefficients for the transverse and longitudinal portions of the field are equal only as a 0. Self-consistency requires that 3 =±1 1 in the case of transverse focal fields and =1 in the case of longitudinal focal fields. A. Transverse Focal Fields Consider now the case of transverse electric and magnetic fields. By using the relation 3 =1 1, where = ±1, Eqs. (33) can be written as so that E 0 =B 0 and E 0 e = B 0 b ẑ e x = b y, e y = b x. 37 38a 38b It is straightforward to calculate the radiant intensity Au using Eq. (31) and the constraint given by Eqs. (38): 1+ cos Au =. 39 coth a cos Notice that this radiant intensity is rotationally symmetric and independent of the directions of ê and bˆ. The total power contained in either field is simply coth a acoth a + coth a. 41 The corresponding directional spread is found to be cos 4 coth a + 3 coth a acoth a 3 coth a + = coth a a csch. a Equations (41) and (4) can be substituted into Eqs. (1) to verify that, in this case, 3 =1 1, as required for consistency. In the paraxial limit a, the measures of spread for the fields with =1 correspond to those of the scalar fields in the same limit. Letting = 1 results in fields that do not approach the paraxial case as a. In fact, the directional spread of such fields increases with increasing a. As will be seen later, the fundamental bounds for the spreads are set only by the =1 case, so the = 1 case may be disregarded; this is intuitive due to the right-hand rule. In the nonparaxial limit a 0, the fields for both the =1 and = 1 cases correspond to crossed electric and magnetic dipoles of equal magnitude and phase located at the origin, which, as found by Sheppard and Larkin [8], achieve the fundamental upper bound of focal energy density obtained by Bassett [9]. Unlike the optical fields that maximize the focal intensity, the mixed-dipole field has some directionality (i.e., /). In fact, as a 0, /3. The energy density, which is rotationally symmetric for these fields, is plotted over the y=0 plane in Fig. 3 for =1 and coth a=1.01, 1.1, and. It can be verified that, for coth a=1.01, which is the nearest to the paraxial limit, the plot of the energy density begins to be similar to the plot of the irradiance for the case of the optical transverse focal field. Note that this field is less rippled than the optical transverse focal field; there the ripples are due to noticeable backward propagating components, which are suppressed here by the null in the = direction of the mixed-dipole distribution. B. Longitudinal Focal Fields In the case where both the electric and magnetic focal fields are longitudinal, the value of the normalization constant is fixed by the constraint =1. Equations (33) then impose no relation between the magnitudes of the electric and magnetic fields. Any linear combination of the two fields is valid. It turns out that, regardless of the ratio E 0 /B 0, the spatial and directional spread measures are identical to those calculated in Eqs. (1) and (). The longitudinal magnetic focal field corresponds to an azimuthally polarized beam, much like the longitudinal electric focal field corresponds to a radially polarized beam.

310 J. Opt. Soc. Am. A / Vol. 4, No. 10 / October 007 Moore et al. Fig. 3. (Color online) Energy density, Er +Br, of an optimal electromagnetic field with transverse polarization =1 at the focus for a given total power E for (a) coth a=1.01, (b) coth a=1.1, and (c) coth a= over the y=0 plane. Note that the focal energy density increases as coth a (and consequently the directional spread of the field) increases. 5. GRAPHICAL REPRESENTATION OF UNCERTAINTY RELATIONS A graphical display of the parametric relation between the focal-irradiance-based spatial spread and the directional spread is presented in Fig. 4 to illustrate the differences among the various cases. As in [1], a modified version of the spatial spread measure that is better suited for graphical purposes is used. Our graphical measure, d R, is defined by d R = arccos D R D R, 43 where D R is the value of D R for a plane-wave amplitude corresponding to a dipole in the optical case and mixed dipoles in the electromagnetic case. The dark gray region in the lower-left corner represents the forbidden region; Fig. 4. Plot of d R versus for the (a) transverse optical, (b) transverse electromagnetic, and (c) longitudinal optical or electromagnetic cases discussed in this paper. In each case, the solid curve represents fields with maximum focal irradiance (energy density), the dotted curve represents the nonparaxial electromagnetic generalization of Gaussian beams referred to here as complex-focus dipole (CFD) fields, and the dashed curve represents fields whose spectra correspond to a dipole truncated by an exit pupil at infinity. In (b), the dotted dashed curve represents the transverse electromagnetic case for which = 1, corresponding to fields with stationary but neither maximal nor minimal focal energy density. The regions occurring below the solid curve are forbidden for all comparable fields (i.e., optical or electromagnetic). i.e., no fields with directional and spatial spreads lying in that region can exist. A comparison of the forbidden regions of the transverse optical case [see Fig. 4(a)] and of the longitudinal optical case [see Fig. 4(c)] clearly shows that the transverse case is more localized than the longitudinal case, and therefore the fundamental lower bound. For the transverse electromagnetic case [see Fig. 4(b)], the fundamental lower

Moore et al. Vol. 4, No. 10/October 007/J. Opt. Soc. Am. A 311 bound does not go to the upper-left corner because the field composed of crossed electric and magnetic dipoles has some directionality [10]. The line segment representing = 1 for the transverse electromagnetic case [represented by a dotted dashed curve in Fig. 4(b)] never approaches the paraxial limit. This line segment could appear to be an upper bound restricting the upper-left corner, but this is not the case because Fig. 4(c) applies to electromagnetic fields as well as optical fields and it can be seen that fields do exist in that region. For the sake of comparison, the relation between the focal-irradiance-based spatial spread and the directional spread for complex-focus dipole (CFD) fields and dipole spectra truncated by an exit pupil (EP) at infinity are included. These fields are chosen because CFD fields are a nonparaxial generalization to Gaussian beams [11,1] and the truncated dipole spectrum is the generalized solution [8] to Luneburg s first apodization problem [13]. The CFD fields (represented by dotted curves in Fig. 4) consist in displacing the dipole(s) to the complex point 0,0,iq and have the following plane-wave amplitude A CFD u;q = U 0 expqu z u ê u + bˆ u. 44 The normalized magnitudes of the plane-wave amplitudes for CFD fields with q=1, 10, and 100 are shown in Fig. 5, as are the normalized amplitudes for fields with maximum focal energy density with coth a=, 1.1, and 1.01, in both the rotationally symmetric optical and electromagnetic cases. Near =0, the angular spectra for these two types of fields exhibit the same behavior when coth a=1+1/q. For increasing values of, the angular spectra of CFD fields approach zero more quickly than those of the fields that maximize the focal energy density. The plane-wave amplitude of the dipole field truncated by an exit pupil at infinity (represented by dashed curves in Fig. 4) is given by 0 u ê u + bˆ u, u z cos M A EP u; M =U 0, u z cos M. 45 For comparison with the optical cases, terms involving bˆ are ignored in Eqs. (44) and (45). The vectors indicating the direction of focal fields ê and bˆ are transverse for Figs. 4(a) and 4(b) and longitudinal for Fig. 4(c). The spreads for these fields are given in Appendix A. 6. CONCLUDING REMARKS Using a variational optimization procedure, the upper bound of the focal energy density of a monochromatic electromagnetic field, given the input power and directional spread, was found. This bound is expressed parametrically (as was also the case for the three-dimensional scalar fields in [1]) and corresponds to fields that have transverse polarization at the focus in both the optical and the electromagnetic cases. The fundamental upper bound on the focal irradiance of a field depends on whether the optical or electromagnetic case is considered. This is due to the fact that an electric dipole field radiates equally in the backward and forward directions, whereas the mixed-dipole field, which maximizes the electromagnetic energy density, is more directional. While the longitudinally polarized focal fields corresponding to radially polarized fields do not result in any fundamental bounds, the spreads for such fields result in a simple relationship between the directional spread and the focal irradiance in the paraxial limit. Fig. 5. Magnitude of the normalized plane-wave amplitude, Au, as a function of the polar angle =arccos u z for CFD fields (dashed curves) and the fields that maximize the focal energy density (solid curves), for the (a) transverse optical and (b) transverse electromagnetic cases. In both cases, the CFD fields are plotted for q=1,10,100 and the fields that maximize focal energy density are plotted for coth a=, 1.1, and 1.01. Notice that the angular spectra of the two field types are very similar for small when coth a=1+1/q but that the angular spectra of CFD fields drop to zero faster than those of the fields that maximize the focal energy density. APPENDIX A: SPREAD CALCULATIONS FOR OTHER FIELDS In this Appendix, expressions for the focal-irradiancebased spatial spread and the directional spread of CFD fields and dipole fields truncated by an exit pupil are given. In order to compare the relationship between the focal energy density and the directional spread of these fields with the corresponding fundamental bounds, these expressions are used, in conjunction with the graphical measure given by Eq. (43), to generate the plots in Section 5. Upon substituting the plane-wave amplitude for a CFD field, given by Eq. (44), into Eqs. (3) and (7) in the optical case (bˆ is ignored), the spreads for a CFD field with a transverse focal electric field ê ẑ=0 can be found to be

31 J. Opt. Soc. Am. A / Vol. 4, No. 10 /October 007 Moore et al. = arccos8q 3 +6qcosh q 8q +3sinh q q4q, +1sinh q q cosh q A1a q 3 4q +1sinh q q cosh q D E R =. A1b 4q +1sinh q q cosh q The same process may be used to find the spreads for a CFD field with a longitudinal focal electric field ê=ẑ: = arccos 6q cosh q + 4q +3sinh q q sinh q +q cosh q, Aa q 3 q cosh q sinh q E or D EM R =. Ab 4q cosh q sinh q In a similar fashion, Eq. (45) can be substituted into Eqs. (3) and (7) in the optical case to find both the spreads for a transverse focal electric field, = arccos 4 3 3 cos M cos 4 M 4 3 cos M cos 3 A3a M, 1 D E R =4 3 cos M cos 3, A3b M and the spreads for a longitudinal focal electric field, = arccos 4 3 1 cos M + cos 4 M 3 cos M + cos 3 A4a M, 6 D E R = 3 cos M + cos 3. A4b M In the electromagnetic case, the spreads are calculated by substitution of the plane-wave amplitude of a field into Eqs. (6) and (8). The plane-wave amplitude for a CFD field, given by Eq. (44), is used to find the spreads for transverse crossed electric and magnetic fields (ê ẑ=bˆ ẑ =0 and ê=bˆ ẑ): = arccos 6q cosh q + 4q +3sinh q q sinh q +q cosh q, A5a q 3 sinh q + 4q qexpq D EM R =. A5b sinh q + qq 1exp q Similarly, Eq. (45) is used to find the spreads of crossed electric and magnetic dipole fields truncated by an exit pupil: = arccos 1 4 17 + 17 cos M + 11 cos M + 3 cos 3 M 7 + 4 cos M + cos M, D R EM =6 1 cos M 7 + 4 cos M + cos M. A6a A6b The spreads for longitudinal focal electromagnetic fields are not included because, as discussed in Subsection 4.B, they do not vary from longitudinal focal fields in the optical case given by Eqs. (A) and (A4). ACKNOWLEDGMENTS Nïcole J. Moore and Miguel A. Alonso acknowledge support from the National Science Foundation through the Career Award PHY-0449708. We also wish to thank Massimo Santarsiero and Riccardo Borghi for several useful discussions and ideas. REFERENCES AND NOTES 1. N. J. Moore, M. A. Alonso, and C. J. R. Sheppard, Monochromatic scalar fields with maximum focal irradiance, J. Opt. Soc. Am. A 4, 057 064 (007).. M. A. Alonso and G. W. Forbes, Uncertainty products for nonparaxial wave fields, J. Opt. Soc. Am. A 17, 391 40 (000). 3. M. A. Alonso, R. Borghi, and M. Santarsiero, Nonparaxial fields with maximum joint spatial-directional localization. I. Scalar case, J. Opt. Soc. Am. A 3, 691 700 (006). 4. These constraints, in conjunction with Eqs. (1), can be used as an additional consistency check on the calculated values of D R E and in each case. 5. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, Focusing light to a tighter spot, Opt. Commun. 179, 1 7 (000). 6. K. S. Youngworth and T. G. Brown, Focusing of high numerical aperture cylindrical-vector beams, Opt. Express 7, 77 87 (000). 7. R. Dorn, S. Quabis, and G. Leuchs, Sharper focus for a radially polarized light beam, Phys. Rev. Lett. 91, 33901 (003). 8. C. J. R. Sheppard and K. G. Larkin, Optimal concentration of electromagnetic radiation, J. Mod. Opt. 41, 1495 1505 (1994). 9. I. M. Bassett, Limit to concentration by focusing, Opt. Acta 33, 79 86 (1986). 10. C. J. R. Sheppard and P. Török, Electromagnetic field in the focal region of an electric dipole wave, Optik (Stuttgart) 104, 175 177 (1997). 11. C. J. R. Sheppard and S. Saghafi, Electric and magnetic dipole beam modes beyond the paraxial approximation, Optik (Stuttgart) 110, 487 491 (1999). 1. C. J. R. Sheppard and S. Saghafi, Transverse-electric and transverse-magnetic beam modes beyond the paraxial approximation, Opt. Lett. 4, 1543 1545 (1999). 13. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, 1964), pp. 386 395.