Focused azimuthally polarized vector beam and spatial magnetic resolution below the diffraction limit

Transcription

1 Research Article Vol. 33, No. 11 / November 2016 / Journal of the Otical Society of America B 2265 Focused azimuthally olarized vector beam and satial magnetic resolution below the diffraction limit MEHDI VEYSI, CANER GUCLU, AND FILIPPO CAPOLINO* Deartment of Electrical Engineering and Comuter Science, University of California, Irvine, California 92697, USA *Corresonding author: f.caolino@uci.edu Received 10 May 2016; revised 31 August 2016; acceted 7 Setember 2016; osted 15 Setember 2016 (Doc. ID ); ublished 11 October 2016 An azimuthally electric-olarized vector beam (APB), with a olarization vortex, has a salient feature that it contains a magnetic-dominant region within which the electric field is ideally null while the longitudinal magnetic field is maximum. Fresnel diffraction theory and lane-wave sectral calculations are alied to quantify field features of such a beam uon focusing through a lens. The diffraction-limited full width at half-maximum (FWHM) of the beam s longitudinal magnetic field intensity rofile and comlementary FWHM of the beam s annular-shaed total electric field intensity rofile are examined at the lens s focal lane as a function of the lens s araxial focal distance. Then, we lace a subwavelength dense dielectric Mie scatterer in the minimum-waist lane of a self-standing converging APB and demonstrate for the first time, to the best of our knowledge, that a very-high-resolution magnetic near-field at otical frequency is achieved with total magnetic near-field FWHM of 0.23λ (i.e., magnetic near-field sot area of 0.04λ 2 ) within a magnetic-dominant region located one radius (0.12λ) away from the scatterer. In articular, the utilization of the nanoshere as a magnetic nanoantenna (so-called magnetic nanorobe) illuminated by a tightly focused APB is instrumental in boosting the hotoinduced magnetic resonse and suressing the electric resonse of a samle matter. The access to the weak hotoinduced magnetic resonse in samle matter would add extra degrees of freedom to future otical hotoinduced force microscoy and sectroscoy systems based on the excitation of hotoinduced magnetic diolar transitions Otical Society of America OCIS codes: ( ) Laser beam characterization; ( ) Electromagnetic otics; ( ) Paraxial wave otics; ( ) Lenses; ( ) Otical vortices; ( ) Near-field microscoy. htt://dx.doi.org/ /josab INTRODUCTION Vector beams [1 7] are a class of otical beams whose olarization rofiles on the transverse lane, erendicular to the beam axis, can be engineered to have an inhomogeneous distribution. Among them, beams with cylindrical symmetry (so-called cylindrical vector beams), articularly radially [3,4,8 10] and azimuthally [11 13] electric-olarized vector beams, are excetionally imortant in the otics community. Owing to the resence of the longitudinal electric field comonent, a radially olarized vector beam with a ring-shaed field rofile after tight focusing through a lens rovides a tighter electric field sot comared to the well-known linearly and circularly olarized beams [8,9]. Such a beam has been extensively examined under tight focusing and has found many rominent alications in article maniulation, high-resolution microscoy, and sectroscoy systems [3,5,6,8,9,14 24]. Here, we are articularly interested in studying the azimuthally electricolarized vector beam rimarily due to its unique magnetic field features: a strong longitudinal magnetic field where the electric field is null. In the following, we denominate such a beam simly as an azimuthally olarized beam (APB) referring to the local orientation of its electric field vector. As schematically shown in Fig. 1, APBs ossess an electric field urely transverse to the beam axis and a strong longitudinal magnetic field comonent in the vicinity of the beam axis where the transverse electric and magnetic fields are negligible and even vanish on the beam axis [12]. This so-called magnetic-dominant region is characterized by the resence of a tight magnetic field with longitudinal olarization. Focusing an APB through a lens boosts its longitudinal magnetic field comonent relatively more than its transverse electric and magnetic fields [12]. Due to such a unique roerty, the APB may be beneficial by adding an extra feature to future sectroscoy and scanning /16/ Journal 2016 Otical Society of America

2 2266 Vol. 33, No. 11 / November 2016 / Journal of the Otical Society of America B Research Article robe microscoy systems [25,26] based on the excitation of magnetic diolar transitions [11,12,27,28]. At otical frequencies, hotoinduced magnetic diolar transitions in matter are weaker than their electric counterarts [28 30] and therefore require an excitation beam with an enhanced magneticdominant region to be exlicitly excited [28]. In this regard, APBs are a very suitable choice for the illumination beam in such sectroscoy and scanning robe microscoy systems. Even though various methods have been roosed to generate APBs [12,31 40], characterization of the magnetic field of these beams under tight focusing, to best of the authors knowledge, remains to be fully elucidated. This study is the basis for the successful imlementation of magnetically sensitive nanorobes at otical frequency, which are crucial in the develoment of magnetism-based sectroscoy alications and the study of weak hotoinduced magnetism in matter [27,41,42]. In this aer, we reort the diffraction-limited tight field (esecially magnetic field) features of an APB, reresented in terms of araxial Laguerre Gaussian (LG) beams, with beam arameter w 0 that is a measure of the satial extent of the beam in the transverse lane at its minimum waist. Two figures of merit are used in this aer to quantify the field features on transverse lanes: (i) the full width at half-maximum (FWHM) of the longitudinal magnetic field intensity and (ii) the comlementary FWHM (CFWHM) of the annularshaed total electric field intensity. Keeing in mind that for a very small beam arameter w 0 the exressions obtained via araxial aroximation may not be accurate, we also reort results using the accurate analytical-numerical lane-wave sectral (PWS) calculation [43], which is analogous to the Richards and Wolf theory [44]. We first elaborate on the diffraction-limited tight focus of an APB through a converging lens using both araxial Fresnel diffraction integral formulation, leading to analytical assessments, and the accurate PWS calculations (see [12] for more details on PWS). We demonstrate using the Fresnel integral under araxial aroximation that uon focusing through a lens an incident APB converts to another APB whose beam arameter is linearly roortional to the lens araxial focal distance and inversely roortional to the incident APB arameter (see Aendix A). The minimum-waist lane osition of the focused beam redicted by the Fresnel integral coincides with the lens araxial focal lane, which deviates from the actual focal lane osition calculated by PWS. The figures of merit of an APB focused by a lens are therefore calculated both by the Fresnel integral at the lens s araxial focus and by the PWS at the actual focal lane as a function of the lens araxial focal distance. In addition to the case of focusing an APB by a lens mentioned above, the tight field features of a self-standing converging APB, as schematically shown in Fig. 1, are also examined, and its figures of merit are calculated using the araxial LG beam exressions and the PWS calculations at the minimum-waist lanes redicted by the resective methods. Recently, it has been exerimentally confirmed that cylindrical vector beams may selectively excite the electric or magnetic diolar resonances of a subwavelength-sized dense dielectric nanoshere (e.g., a silicon nanoshere) [24]. In this aer, we use a silicon nanoshere as a magnetic nanoantenna (so-called Fig. 1. Schematic of a converging azimuthally electric-olarized vector beam (APB), with a longitudinal magnetic field on its axis. magnetic nanorobe) and lace it at the focus of a converging APB that selectively excites a magnetic diolar resonance in the nanoshere as in [24]. The aim is to achieve a subwavelength magnetic field resolution. In general, such a subwavelengthsized scatterer hosts a magnetic Mie resonance with a circulating electric dislacement current in addition to an electric diolar resonance. However, the latter is not excited by an APB due to its cylindrical symmetry, which ideally leads to a null average dislacement current over the nanoshere. The induced electric dislacement currents with a net magnetic diole moment in the Si nanoshere along the z direction are shown to boost not only the total longitudinal magnetic field but also the satial magnetic field resolution below the diffraction limit in the vicinity of the scatterer. A total magnetic field enhancement of about 2.3 (with resect to the total incident magnetic field) and a total magnetic field sot area as small as 0.04 λ 2 are achieved within a magnetic-dominant region, evaluated at a transverse lane one nanoshere radius (0.12 λ) away from the scatter surface. Note that FWHM is an effective feature to characterize the magnetic near-field intensity. The FHWM is also here used as a shorthand measure of resolution, i.e., the minimum resolvable distance between two closely saced oint sources because the side-lobe eak of the magnetic near-field intensity rofile is for all cases by far less than half of its main eak. Throughout the aer, we also consider time harmonic fields with an ex iωt time deendence, which is suressed for convenience. Furthermore, bold symbols denote vectors and hats (^) indicate unit vectors. 2. CHARACTERIZATION OF AN APB APB is here exressed as a suerosition of a left- and a righthand circularly olarized beam, carrying orbital angular momentum (OAM) with orders of 1 and 1, resectively. In araxial regimes, OAM-carrying beams are analytically reresented as LG beams [1]. Thus, the APB s electric field is exressed in terms of self-standing araxial LG beams in cylindrical coordinate system as [12] E i ffiffiffi 2 u 2 1;0 ê RH u 1;0 ê LH e ikz ; (1) where ê RH ˆx iŷ ffiffiffi 2 and êlh ˆx iŷ ffiffiffi 2 are, resectively, right- and left-hand circularly olarized unit vectors, and the LG beam exression is

3 Research Article Vol. 33, No. 11 / November 2016 / Journal of the Otical Society of America B 2267 u 1; 0 V ffiffiffi 2ρ π w 2 e ρ w 2ζ e 2i tan 1 z z R e iφ ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w w 0 1 z z R 2 ; ζ 1 iz z R ; (2) where V is an amlitude coefficient (in volts), z R πw 2 0 λ is the Rayleigh range, and k 2π λ and λ are the wavenumber and wavelength in the host medium, resectively. The beam arameter w 0 controls the transverse satial extent of the beam at its minimum-waist lane. Vaguely seaking, w 0 corresonds to the minimum waist, which is very well defined for the fundamental Gaussian beam (FGB). Since the actual waist of the APB differs from w 0, we refer to call it simly as beam arameter because this difference is of relevance in this aer. Here, the term beam waist is reserved for the minimum of the actual waist size as discussed next. The electric field in Eq. (1) is equivalently exressed as [12] E E φ ˆφ V ffiffiffi 2ρ π w 2 e ρ w 2ζ e 2i tan 1 z z R e ikz ˆφ; (3) which clearly shows the urely azimuthal olarization of the beam. The electric field intensity rofile of an APB is lotted at the beam s minimum-waist lane (i.e., z 0) in Fig. 2(a). It is observed that the APB s electric field has an annular-shaed intensity rofile whose CFWHM is of interest to us as a measure of the beam s tightness. The APB examined in Fig. 2 is carrying a ower of 1 mw, which is obtained by setting V 0.89 V in Eq. (3) and its beam arameter is set to w λ. In Aendix B and Section 4 of this aer, we show that the converging APB exressed by Eq. (1) with such an illustrative beam arameter (w λ) reresents, by a good aroximation, a self-standing beam. The strength of the APB s electric field given in Eq. (3) is roortional to ρ ex ρ 2 w 2 in any given z transverse lane, and it reaches its maximum je φ ρ M ;z j jv ffiffiffi j 2ρ ffiffiffi M π w 2 e ρ M w 2 2 ffiffiffiffiffi jv j (4) πe w at ρ M w ffiffiffi 2. Therefore, on the minimum-waist lane (i.e., z 0) the electric field magnitude eaks at ρ M w 0 ffiffiffi 2, that is in an agreement with what is shown in Fig. 2(a). The magnetic field of the APB with the electric field given in Eq. (3) is subsequently found by using iωμh E in cylindrical coordinates, yielding a longitudinal magnetic field comonent as [12] Fig. 2. Intensity rofile of (a) the electric field, (b) the axis-confined longitudinal magnetic field, and (c) the urely radial transverse magnetic field for an APB carrying 1 mw ower and with beam arameter of w λ at λ 523 nm. H z V ffiffiffi 4i ρ 2ζ π w 2 1 e ρw 2ζ e 2i tan 1 z z R e ikz (5) ωμ w alongside a radial magnetic field comonent as H ρ 1 η E φ 1 1 ρ 2 2w 2 0 kz R w 2 : (6) It is observed from Eq. (6) that for kz R 1 the radial magnetic field comonent follows the electric field rofile of the beam. In summary, the APB ossesses only E φ, H z, and H ρ field comonents. The intensity of the APB s longitudinal magnetic field [given in Eq. (5)] is lotted in Fig. 2(b), where it eaks on the beam axis (ρ 0) and is characterized by its FWHM. The maximum of the longitudinal magnetic field strength at any z is given by jh z ρ 0;z j 4jV j w 2 ωμ ffiffiffi π (7) and is thus inversely roortional to w 2. It is observed from Eq. (7) that the longitudinal magnetic field of the APB eaks at the beam s minimum-waist lane (i.e., z 0, where w w 0 ), where its magnitude is inversely roortional to the square of the beam arameter w 2 0. The transverse magnetic field [which is urely radial and given in Eq. (6)] increases together with the electric field (which is urely azimuthal) as the radial distance ρ from the beam axis increases and eaks away from the beam axis alongside the azimuthal electric field, as shown in Fig. 2(c). By duality, this is analogous to the case of the radially olarized beam in which electric field intensity is urely longitudinal on the beam axis and its transverse comonent eaks off the beam axis [3,4,6,11]. Here, we define the CFWHM for the annular-shaed electric field intensity rofile of the APB as the width across its null, where the field intensity rises to the half of its maximum, i.e., to 0.5jE φ ρ M ;z j 2 [see Eq. (4) and Fig. 2(a)]. In addition, the FWHM of the longitudinal magnetic field intensity is also calculated as the width across its eak on the beam axis, where the longitudinal magnetic field intensity dros to the half of its maximum, i.e., to 0.5jH z ρ 0;z j 2 [see Fig. 2(b)]. Based on the azimuthally olarized electric field and the longitudinally olarized magnetic field exressions given, resectively, in Eqs. (3) and (5), the CFWHM of the electric field intensity and the FWHM of the longitudinal magnetic field intensity at the minimum-waist lane are calculated and given by CFWHM E φ j z w 0 ; FWHM H z j z w 0 : (8) One may be also interested in the ratio of the longitudinal magnetic field on the beam axis, where it is maximum, to the maximum of the electric field at ρ ρ M. This ratio, normalized with resect to the inverse of the host-medium wave imedance η 1 ffiffiffiffiffiffiffi ε μ, is equal to η jh ffiffi z ρ 0;z j 2 λ je φ ρ M ;z j π w e λ w : (9) Note that such ratio is inversely roortional to w, and it reaches its maximum at z 0, i.e., in the minimum-waist lane. Therefore, the maximum magnitude of the longitudinal magnetic field increases relatively more than the maximum

4 2268 Vol. 33, No. 11 / November 2016 / Journal of the Otical Society of America B Research Article magnitude of the electric field as w 0 decreases (tighter beams). Note that decreasing w 0 also has the effect of decreasing the area of the longitudinal magnetic field sot. Finally, we should note that on the minimum-waist lane (z 0) the ratio ηjh z ρ; 0 j je φ ρ; 0 j is equal to unity at the radial distance ρ w 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi πw λ πw 0 2λ! (10) and inside this radius, the longitudinal magnetic to total electric field contrast ratio for APB is larger than the magnetic to electric field contrast ratio (the admittance) of a lane wave 1 η. The otical ower carried by the APB is calculated by the integral of its longitudinal Poynting vector over its minimum-waist lane as Z 2π Z P 1 Ref E 2 φ H ρ g z 0 ρdρdφ: (11) 0 0 After substituting the APB s azimuthal electric and radial magnetic fields formulas given in Eqs. (3) and (6) into Eq. (11), the ower carried by the APB is evaluated as R jv j π R 0 je φ j 2ρ 2 0 w 0 jv j 3! dρdφ P 1 2η πw 0 λ 2 : (12) After a change of variable from 2 ρ w 0 2 to t, the integral term in Eq. (12) is found to be equal to 0.5Γ 3 1, where Γ is the gamma function. Therefore, Eq. (12) is reduced to P jv j2 2η πw 0 λ 2 : (13) Equation (13) clearly shows that ower carried by the APB is exlicitly exressed as a function of w 0 λ and the absolute value of the amlitude coefficient jv j. Therefore, the APB s amlitude coefficient V is obtained for certain beam arameter w 0 and required ower using Eq. (13). In order to have a better assessment of the APB s significance in roviding a magnetic-dominant region, here we comare an APB with a traditional FGB of equal owers and beam arameters. The electric and magnetic field distributions of a araxial APB and a FGB at their araxial minimum-waist lanes, i.e., at z 0, are comared in Fig. 3 for two illustrative beam arameters set to (to row) w 0 0.9λ and (bottom row) w 0 0.5λ. Here, the APB and the FGB carry equal owers of 1 mw [see Eq. (13)]. In order to have azimuthally symmetric magnetic field distribution for the FGB, we consider circularly olarized FGB in Fig. 3; however, similar conclusions would be obtained if we used linearly olarized FGB. In contrast to the FGB whose magnetic and electric fields eak on the beam axis (i.e., the z axis), the APB contains a ure longitudinal magnetic field comonent on the beam axis where its electric field vanishes. The magnetic-to-electric field intensity ratio normalized to that of a lane wave is also lotted for the APB and the FGB in Fig. 3 (third column) varying radial distance from the beam axis. Note that the magnetic-to-electric field intensity ratio of the FGB is very close to that of a lane wave. In contrast, the APB has a very large magnetic-to-electric field intensity ratio in Fig. 3. Comarison between a FGB and an APB of equal owers (1 mw) at λ 523 nm and beam arameters at their minimum-waist lanes (i.e., z 0): (to row) w 0 0.9λ and (bottom row) w 0 0.5λ. Strength of the total electric field (first column), strength of the total magnetic field (second column), and the ratio of the total magnetic to the total electric field intensities normalized to that of a lane wave (third column). (Note how this ratio grows for the APB when aroaching the beam axis.) the vicinity of the beam axis denoting the magnetic-dominant region. This ratio for the APB tends to infinity when ρ 0. For w 0 0.9λ [Fig. 3 (to row)], even though the strength of the APB s magnetic field on the beam axis is half of that of the FGB carrying the same ower, the APB uniquely has only magnetic field and no electric field there, which is an imortant feature that can be used in various alications. In addition, it is observed from Fig. 3 (to row) that the FWHM of the total magnetic field for the APB with w 0 0.9λ is larger than that for the FGB with the same w 0 ; this is attributed to the fact that the APB contains an annular-shaed radial magnetic field comonent [see Fig. 2(c)]. However, decreasing the beam arameter (tightening the beam) from w 0 0.9λ to 0.5λ boosts the longitudinal magnetic field comonent relatively more than the radial one and therefore significantly decreases the FWHM of the total magnetic field, as shown in Fig. 3 (bottom row). To further reduce the FWHM of the total magnetic field of the APB aroaching that of its longitudinal magnetic field comonent, one aroach might be to use ring-shaed lenses with high numerical aertures for focusing of the APB. This technique has been used for generating very shar electric field focuses using radially olarized beams [8,9]. 3. FOCUSING AN APB THROUGH A LENS In this section, we aim at characterizing magnetic and electric fields of an APB at the focal lane of a lens through which the APB is focused. To have first an analytical assessment, we use the Fresnel integral as summarized in Aendix A and calculate the fields at the lens s araxial focal lane. We show in Aendix A that a lens, under araxial aroximation, converts an incident APB, whose minimum-waist lane occurs at the lens surface, to another converging self-standing APB whose araxial minimum-waist lane coincides with the lens s

5 Research Article Vol. 33, No. 11 / November 2016 / Journal of the Otical Society of America B 2269 Fig. 4. Schematic of a converging lens transforming an incident APB with beam arameter w 0;i into another converging APB with beam arameter w 0;f. The magnitudes of the total electric (which is urely azimuthal) and the longitudinal magnetic fields are lotted. The radial comonent of the magnetic field, also exeriencing focusing, is not shown here for brevity. In this reresentative examle, the incident APB carries 1 mw ower, and the lens radius and focal distance are set at a 40λ and f 80λ, resectively. The beam arameters of the incident and focusing APBs are w 0;i 29λ and w 0;f 1.3λ, resectively. araxial focal lane. Results are schematically reresented in Fig. 4 for a secific examle where we show the total electric and the longitudinal magnetic field magnitudes of the APB before and after focusing through the lens. Next, in order to confirm the analytical calculations and rovide a guide to where the Fresnel integral exressions are accurate, we characterize the APB uon focusing through a converging lens using accurate PWS calculations. As for the PWS calculations, we assume the thin lens aroximation such that each ray entering one side of the lens exits the other side at the same transverse (ρ, φ) coordinates as the entrance osition. We model the transmission through the lens by imosing a hase shift, which varies in the radial direction, added to the ρ-deendent hase of the incident APB. The transmission hase shift that is added, relative to a sherically converging wave, is given by Φ ρ 2πf λ sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ρ2 f 2! 1 ; (14) where f is the araxial focal distance of the lens (on the right anel of Fig. 4), ρ is the local radial coordinate of the lens, and λ is the wavelength in the host medium on the right side of the lens. Note that we are assuming that the lens does not vary the ρ-deendent amlitude of the incident APB s field across the lens. We also must remember that in the Fresnel integral equation, the hase term in Eq. (14) is araxially aroximated as a quadratic hase [see Eq. (A2)] term [45]. The Fourier and inverse Fourier transform integrals in PWS calculations (see Eqs. (24) and (25) in [12]) are then numerically calculated via a two-dimensional FFT algorithm, where the satial domain size and the satial resolution are 102.4λ 102.4λ and λ 20, Fig. 5. (a) FWHM of the longitudinal magnetic field intensity jh z j 2, and (b) CFWHM of the annular-shaed electric field intensity jej 2 calculated using (i) PWS at the actual focal lane and (ii) the Fresnel integral (FI) at the lens s araxial focal lane, uon illuminating the lens by an incident APB, varying the normalized lens s focal distance f. resectively. The corresonding sectral domain size and the sectral resolution are 20k 20k and 0.01k, resectively, extending to evanescent sectral comonents as big as 10k. Moreover, to model the hard-edged aerture, the electric field is assumed null outside of the overall lens aerture in the lens lane. We now characterize the FWHM of jh z j 2 and the CFWHM of jej 2 at the lens focal lane for an incident APB. As a reresentative examle, we set the lens radius a equal to 40λ and characterize the focusing beam at the lens s focal lane as the lens s araxial focal distance f changes. The incident APB has a beam arameter of w 0;i 29λ such that the beam cross section is much wider than the wavelength and 90% of the incident beam ower illuminates the lens surface. In Fig. 5, we lot the FWHM of jh z j 2 and CFWHM of jej 2 calculated at the lens s focal lane as a function of the lens radius to focal distance ratio a f, where a is ket constant and f is varied. We recall that the right side of Fig. 4 corresonds to the field mas for a secific case with a f 2, which is a oint on the curves reorted in Fig. 5. The quantities lotted in Fig. 5 are calculated using both the Fresnel integral formula [given in Eq. (A10)] at the lens s araxial focal lane (z f ) and PWS calculations (refer to [12] for more details on PWS) at the lens s actual focal lane. It is observed from Fig. 5 that the Fresnel integral results (denoted by FI) agree very well with the accurate PWS results, esecially for large focal distances (small a f ). We also observe from the PWS results that for the case with f a, the FWHM of the longitudinal magnetic field intensity and CFWHM of the total electric field intensity at the lens actual focal lane are 0.715λ and 0.53λ, resectively. Note that the actual focal lane, obtained from PWS calculations, is slightly dislaced from the lens s araxial focal lane as described in the next section. In Aendix B, we elaborate more on this as we examine the lane-wave sectrum of converging beams. 4. SELF-STANDING CONVERGING APB As we discussed in the revious section, a lens transforms an incident APB to another converging self-standing APB (see Fig. 4 and Aendix A). Such transformation simlifies the calculations of the focusing beam due to a lens, assuming

6 2270 Vol. 33, No. 11 / November 2016 / Journal of the Otical Society of America B Research Article the araxial aroximation. Therefore, in this section, we examine roagation of a self-standing converging APB, assuming it is focused by the lens, and quantify its roerties at its minimum-waist lane. We show some imortant tight field features of self-standing converging APBs as a function of their beam arameter w 0;f (as shown in Fig. 4), where we ay articular attention to the FWHM of their longitudinal magnetic fields at the minimum waist. Since we only elaborate on a self-standing converging beam, we dro the subscrit f and denote the converging beam arameter simly as w 0. Note that, in general, the evanescent comonents of the field transmitted through lens are negligible and filtered out as the beam roagates toward the focal lane. The sectral comonents of the converging APBs [exressed by Eq. (1)] are examined in Aendix B, where we show that more than 95% of the sectral energy of the converging APBs with w 0 0.5λ is confined in the roagating sectrum. Thus, in the subsequent studies, the beam arameter of the converging APB is set larger or equal to 0.5λ. The results ertaining to the araxial beam roagation are also comared to those obtained from the analyticalnumerical comutation based on the PWS. We assume to know the initial APB s field distribution, with converging features, at a certain z lane (the so-called reference lane) and observe the beam roagating toward its minimum-waist lane in z direction. In other words, we investigate the converging roerties of the beam on the right side of the lens in Fig. 4. We first assess the validity of the araxial aroximation for APBs as in Eq. (1). It is known that the araxial aroximation for a beam holds under the following condition [2,46,47]: ψ 2k z 2 ψ z 2 ; (15) where E ψe ikz reresents araxial field distribution for a beam roagating in the z direction. In order to determine the validity range of the araxial field, we define a araxiality figure as F ψ 2k z 2 ψ z 2 ; (16) which is a function of local coordinates. We also define the normalized weighted average figure of the araxiality at each transverse z lane as R R F jψj 2 dxdy F ;ave R R jψj 2 dxdy ; (17) where the numerator is the average araxiality figure weighted by the intensity of the transverse field, and the denominator is the total weight of the transverse field intensity with resect to which we normalize the weighted average araxiality figure. The value of F ;ave for the araxial APB s electric field in Eq. (1) is calculated and lotted in Fig. 6 as a function of the beam arameter w 0 at the beam s araxial minimum-waist lane z 0. The larger the araxiality figure F ;ave is, the better the araxial aroximation is. It is observed from Fig. 6 that F ;ave 50 [i.e., log 10 F ;ave 1.7] for beam arameters larger than 0.9λ. We assume that F ;ave values larger than 50 reresent reasonably valid araxial beams for ractical uroses. Thus, for such values of w 0, the araxial electric field Fig. 6. Normalized weighted average figure of araxiality F ;ave (in logarithmic scale) for a converging APB at the beam s araxial minimum-waist lane (z 0) as a function of the beam arameter w 0. exression given in Eq. (1) reresents a self-standing APB s field distribution with a good aroximation. Remarkably, the signature of this validity range manifests itself in the comarison of the araxial beam roagation and the accurate PWS results discussed in the following. We now examine the magnetic and electric field features of a self-standing converging APB at its minimum-waist lane as a function of the beam arameter w 0 (for w 0 0.5λ). With this in mind, we characterize self-standing converging APB using PWS calculations. We start with an APB s araxial transverse field distribution on a transverse reference lane located at z z r (z r <z f ; see Fig. 1 for z f )givenbyeq.(1). Subsequently, the evolutions of the beam s magnetic and electric fields in the ositive z direction are examined using the PWS calculations. The location of the actual minimum-waist lane of a converging APB launched from a reference lane at z r 3.5λ with the field distribution given in Eq. (1) is calculated using PWS and lotted in Fig. 7 as a function of the beam arameter w 0.Weobserve that the actual minimum-waist lane of the converging APB does not occur at z 0, that is the location of the focus redicted by the araxial field exression. This difference is attributed to the resence of lane-wave constituents with large transverse wavenumbers in the field sectrum of the converging APB, which are not roerly modeled in the araxial field exressions (see Aendix B for more details on the sectral content of the APB). For an APB, with decreasing w 0 a larger amount of constitutive roagating lane-wave sectral comonents of the beam s field will have large transverse wavenumbers. Fig. 7. Actual osition of the minimum-waist lane of the beam z f as a function of the beam arameter w 0 for both APB and FGB (calculated using the PWS). The minimum-waist lane estimated by using the simle araxial field exression is at z 0.

7 Research Article Vol. 33, No. 11 / November 2016 / Journal of the Otical Society of America B 2271 Fig. 8. PWS and araxial calculations for (a) the FWHM of the longitudinal magnetic field intensity and (b) the CFWHM of the annular-shaed electric field intensity of the converging APB as a function of the beam arameter w 0. Hence, the difference between the actual minimum-waist lane s location (here, denoted by z f ) and the one redicated by the araxial field exressions (here at z 0) becomes more significant because of the loss of accuracy of the araxial aroximation with decreasing w 0. Such a difference is not unique to the APBs, and a similar difference is observed for traditional FGBs, as shown in Fig. 7. We observe that the difference between the actual and the araxial minimum-waist lane s ositions for a FGB is also increasing as w 0 decreases. However, the difference between the actual minimum-waist lane s osition and the araxial one is larger for an APB than for a FGB with an equal w 0. This is due to the fact that the transverse wavenumber sectrum of an APB s field is broader than that of a FGB s field with the same beam arameter; hence, the araxial aroximation is coarser for the APB comared to the FGB. Next, the FWHM of jh z j 2 and the CFWHM of jej 2 of the converging APB are calculated using both araxial and PWS calculations and lotted in Fig. 8. The FWHM and the CFWHM in the PWS calculations are evaluated at the actual minimum-waist lane of the APB (z z f ), which deends on w 0 (see Fig. 7). Instead, the FWHM and CFWHM under the araxial aroximation are evaluated using Eq. (1)atz 0 for all the w 0 cases. It is observed from the araxial calculations that the FWHM and CFWHM curves decrease monotonically as the beam arameter w 0 decreases. However, in ractice, the decrease in FWHM of the longitudinal magnetic field intensity rofile as well as CFWHM of the electric field intensity rofile is hamered by an ultimate limit imosed by the diffraction of the beam. It is observed from the PWS curves in Fig. 8 that the FWHM and CFWHM of the converging APB are saturated by the diffraction to about 0.56λ and 0.43λ, resectively, desite that the araxial aroximation estimates much smaller FWHM and CFWHM. Thus, according to accurate PWS calculations, a longitudinal magnetic field intensity rofile with FWHM as small as 0.56λ (sot area of about 0.25λ 2 ) is achievable with w 0 0.5λ. The sot area is defined here as the circular area whose diameter is equal to the FWHM. Here, based on what is discussed in Aendix B, we stress that the transverse wavenumber sectrum of the APB s field in Eq. (1) with very small w 0 (w 0 < 0.5λ) is not confined only in the roagating wavenumber sectrum, and it starts to extend to the evanescent sectral region and therefore is not shown in Fig. 8. However, the satial field distribution in Eq. (1) with w 0 as small as 0.5λ has wavenumber sectral constituents still confined in the roagating sectrum (see Aendix B), and it has a relatively large normalized weighted average figure of araxiality F ;ave 14. Therefore, though it may not reresent a strictly self-standing APB, the araxial aroximation is not too coarse. When the beam arameter w 0 is larger than 0.9λ, the araxial curves for the FWHM and the CFWHM in Fig. 8 follow the accurate PWS ones very well, and they start to deviate from PWS curves when w 0 decreases to smaller values, which is in agreement with our finding in Fig. 6. In order to clarify the effect of beam arameter w 0 on different magnetic field comonents of the APB, in Fig. 9 we lot the strength of the longitudinal (H z ) and the radial (H ρ ) magnetic field comonents as well as the strength of the azimuthal electric field normalized to the host-medium wave imedance for two illustrative w 0 values at λ 523 nm, using PWS calculations. We recall that the APB has a H z rofile that eaks on the beam axis (ρ 0), whereas its transverse magnetic field comonent is urely radial and eaks off the beam axis. It is observed from Fig. 9 that the longitudinal magnetic field sot areas as small as 0.25λ 2 and 0.49λ 2 are obtained with converging APBs with w 0 of 0.5λ and 0.9λ, resectively. However, since APB ossesses a radial magnetic field comonent over an annular-shaed region in addition to the longitudinal one (see Fig. 9), the FWHM of the total magnetic field is always larger than the FWHM of the longitudinal magnetic field comonent. It is also observed from Fig. 9 that when w 0 of the APB decreases from 0.9λ to 0.5λ, the strength of its longitudinal magnetic field comonent increases by about 2.2 times, which is relatively more than the increase in the strength of its radial magnetic field comonent (1.24 times). Indeed, as the beam arameter w 0 decreases, the lane-wave sectral distribution of the APB includes large transverse wavenumbers. For smaller beam arameters, such as w 0 0.5λ, a larger ortion of the constitutive transverse electric (TE, with resect to z) lane waves in the sectrum of the APB ossess large transverse wavenumbers, meaning that they roagate in directions with larger angles α with resect to the beam axis, as shown in Fig. 10. Therefore, the magnetic fields of the TE constitutive lane waves, which are erendicular to the lane-wave roagation directions, are more aligned with the beam axis. The deth of Fig. 9. Strength of the longitudinal (H z ) and radial (H ρ ) magnetic fields of an APB for two different beam arameters w 0 at λ 523 nm, evaluated using accurate PWS calculations. The strength of the azimuthal electric field (E φ ) normalized to the wave imedance is also lotted for comarison.

8 2272 Vol. 33, No. 11 / November 2016 / Journal of the Otical Society of America B Research Article Fig. 10. Raytracing model of an APB focusing through a lens. Magnetic field vectors are denoted by blue arrows. Sectral comonents with large transverse wavenumbers rovide strong longitudinal magnetic fields. Fig. 12. Schematic of a converging APB with w 0 0.9λ illuminating a subwavelength-size silicon nanoshere (as a magnetic nanorobe) laced at the actual minimum-waist lane of the beam: r 62 nm, z f 0.55λ, and λ 523 nm. Fig. 11. DOF of the longitudinal magnetic field intensity rofile for a converging APB as a function of the beam arameter w 0, evaluated using PWS. For comarison, the deth of focus of the electric field intensity rofile of a FGB is also lotted. focus (DOF or axial FWHM) of the longitudinal magnetic field intensity rofile for a converging APB is also shown in Fig. 11 as a function of w 0, using accurate PWS calculations. For the sake of comarison, we also lot the DOF of the electric field intensity rofile for a traditional circularly olarized FGB in Fig. 11. Asw 0 increases, the Rayleigh range z R increases as w 0 2 and as a result, the beam waist w varies less with resect to z [see Eq. (2), where w is written as a function of z and z R ). Therefore, the DOF is much longer for larger w 0, which also means the field features are less tight. 5. SPATIAL MAGNETIC RESOLUTION BELOW THE DIFFRACTION LIMIT So far, we have demonstrated that fields in the focal lane of a converging APB with w 0 0.5λ are constructed only from the roagating sectrum and therefore they are diffraction limited (see also Aendix B). We have shown using PWS calculations in Fig. 8 that the transverse FWHM of the longitudinal magnetic field intensity and the CFWHM of the total electric field intensity for such a converging APB are limited by diffraction to 0.56λ and 0.43λ, resectively. In addition, we have also shown that the total magnetic field intensity is less collimated than the longitudinal magnetic field due to the resence of the strong annular-shaed transverse magnetic field. In this section, we aim at enhancing the longitudinal magnetic field of an APB and boosting its satial magnetic resolution below the diffraction limit. To overcome the diffraction barrier, evanescent waves should be excited. One oular aroach to generate evanescent waves, required for achieving satial resolutions below the diffraction limit in microscoy, is to use a subwavelength scatterer [48]. Here, we show that a suer-tight magnetic-dominant sot is achieved using a subwavelength-size dense dielectric Mie scatterer (here, silicon nanoshere) having a magnetic Mie resonance as a magnetic nanorobe [41,42]. We adot an initial araxial electric field distribution for the APB at a reference lane z z r (here, z r 3.5λ) away from the minimumwaist lane based on Eq. (1), as shown in Fig. 12. Recall that based on Aendix B, Figs. 6 and 8 and their corresonding discussions in the aer, the APBs with beam arameters larger than w 0 0.9λ can be, by a good aroximation, reresented by Eq. (1). Therefore, we adot w 0 0.9λ for the illuminating APB to have a tight magnetic field sot. In the revious section, the roagation of such an APB was modeled using the PWS and its accurate minimum-waist lane osition and field distributions are calculated. Here, we imort the APB s araxial transverse electric field distribution [given by Eq. (1)] into the finite integration technique in the time-domain solver imlemented in CST Microwave Studio as a boundary field source. As a consequence of the Schelkunoff equivalence rincile (PEC equivalent) imlemented in CST Microwave Studio, the APB roagates toward the z direction in Fig. 12. The coefficient V in Eq. (2) is set to 0.89 V such that the total ower of the incident APB given in Eq. (13) is 1 mw. In the full-wave simulations, we assume a free-sace wavelength of λ 523 nm. The magnetic field ma of the incident APB (without any scatterer yet), calculated by the time-domain solver imlemented in CST Microwave Studio in the y z longitudinal lane, is shown in Fig. 13(a). We observe from Fig. 13(a) that the APB s minimum-waist lane occurs at z f 0.55λ, which is also obtained by the PWS calculations according to Fig. 7 (the araxial aroximation instead would estimate a focus at z 0). Next, a subwavelength-size silicon nanoshere scatterer is laced at the APB s actual minimumwaist lane (z z f 0.55λ), which is assumed to be in vacuum. The silicon nanoshere has a relative ermittivity equal to ε r 17.1 i0.084 and radius of r 62 nm such that its magnetic Mie olarizability magnitude eaks at λ 523 nm [27]. The total magnetic field s magnitude in the resence of the nanoshere (suerosition of incident and scattered fields)

9 Research Article Vol. 33, No. 11 / November 2016 / Journal of the Otical Society of America B 2273 Fig. 13. Full-wave simulation results for the magnitude of (a) the incident magnetic field and (b) the total magnetic field (summation of incident and scattered field from the nanoshere) locally normalized to the incident magnetic field. locally normalized to the magnitude of the incident magnetic field is also shown in the y z longitudinal lane in Fig. 13(b), where we observe large magnetic field enhancement at the scatterer cross section and in the vicinity of the scatterer. Note that the enhanced magnetic field is strongly localized close to the scatterer with relatively very low side-lobe levels resulting in a very high satial magnetic resolution. Moreover, starting from the surface of the scatterer, the tight magnetic sot extends into the surroundings and dros raidly away from the nanoshere s surface, revealing the resence of evanescent sectral fields in the near-field close to the scatterer. The normalized magnetic and electric field intensities without the resence of the scatterer (only the incident APB) at the APB s minimum-waist lane (at z z f ) and with the scatterer at two different x-y transverse lanes (at z z f r and z z f 2r) slightly away from the scatterer are also shown in Figs. 14(a) and 14(b). We observe from Fig. 14(a) that a high-resolution magnetic field is obtained with a total magnetic near-field intensity with FWHM of 0.108λ (0.23λ) and a side-lobe eak of (0.29), relative to the maximum, at the transverse lane tangent to the shere at z z f r (at the transverse lane one radius away from the shere edge at z z f 2r). Note that the increase of the relative side-lobe level with the distance z from the nanoshere surface is attributed to the increase of the annularshaed radial magnetic field strength relative to the longitudinal magnetic field one (similar trend as in Fig. 9). Figure 14(c) also shows the FWHMs of the total and the longitudinal magnetic near-field intensities on x y transverse lanes with z>z f. The longitudinal magnetic field remains well collimated with no side lobes, whereas the total magnetic field starts to have significant side lobes as the distance z from the nanoshere surface increases because of the growth of the radial magnetic field comonent. Therefore, the FWHM of the total magnetic field intensity is not reorted in Fig. 14(c) for that z range (i.e., z z f > 2.5r) where the FWHM would not be a measure of resolution anymore. The total magnetic field sot areas reorted here (i.e., 0.009λ 2 and 0.04λ 2 ) are much smaller than the ultimate sot area obtained for the longitudinal magnetic field of a tightly focused APB with w 0 0.5λ without the nanoshere, which is 0.25λ 2. The enhancement of the total magnetic field with resect to that of the incident APB at two different transverse lanes is also lotted in Fig. 15(a), where we observe a significant enhancement of the total magnetic field close to the nanoshere. The longitudinal magnetic field of the incident APB induces a magnetic diole moment in the nanoshere olarized along the z direction, which in turn boosts the total magnetic field thanks to the diolar magnetic near-fields. The square of the near-field admittance, defined as the total magnetic field intensity divided by the total electric field intensity, normalized to that of a lane wave (1 η 2 ) is also lotted in Fig. 15(b), which clearly shows a very high contrast ratio between magnetic and electric fields, esecially around the beam axis. On the beam axis (ρ 0) where the electric field has a null, the magnetic to electric field contrast ratio goes to infinity (not shown here). The utilization of a silicon nanoshere as a magnetic nanorobe excited by a converging APB, which rovides a tight Fig. 14. Normalized total (a) magnetic and (b) electric near-field intensities (each case is normalized to its own maximum) without (black solid curves) and with (blue dashed and red dotted curves) the resence of the silicon nanoshere centered at z z f, evaluated at different x-y transverse lanes. (c) The FWHM of the total and the longitudinal magnetic near-field intensity atterns at different transverse lanes with z>z f. Fig. 15. (a) Total magnetic field (summation of incident and scattered fields) of the scatter system locally normalized to that without the nanoshere at z z f, evaluated at different transverse lanes away from the scatterer. (b) Ratio of the total magnetic field intensity to the total electric field intensity of the scatter system normalized to that of lane wave (this defines the local near-field admittance normalized to that of the lane wave).

10 2274 Vol. 33, No. 11 / November 2016 / Journal of the Otical Society of America B Research Article magnetic-dominant region with enhanced magnetic and negligible electric near-fields, is esecially crucial for the exlicit excitation of hotoinduced magnetic diolar transitions in samle matter, with nanoscale recision. Magnetic diolar transitions are, in general, weak when excited at otical frequencies and overshadowed by their stronger electric counterarts. Note that when the nanoshere is utilized as a magnetic nanorobe that boosts the magnetic resonse of a samle matter in its close vicinity, the beam axis and the nanoshere should be erfectly aligned in order to boost selectively the magnetic resonse of the samle matter. This would be advantageous in future scanning robe microscoy based on hotoinduced magnetism as the extreme sensitivity to alignment would result in highresolution maing. In addition, in ractical alications, the nanoshere is either deosited on to of a substrate or located inside a host. Most imortantly, the resence of the dielectric/substrate would (i) shift the focal lane of the converging APB, which could be comensated in a design rocedure when known, and (ii) change the transverse FWHM of the longitudinal magnetic field and the CFWHM of the total electric field inside the dielectric medium. Note that the electric field [which is urely transverse and given in Eq. (3)] and also the magnetic field of the APB should be continuous across the dielectric surface according to the field boundary conditions. However, since the effective wavelength inside the dielectric is shorter than that in the vacuum, the field s features of the illuminating APB would be tighter than those belonging to the same APB roagating inside unbounded vacuum. Nevertheless, the magnetic-dominant region would be reserved across the dielectric surface. The effect of the dielectric interface on electric field evolution of tightly focused azimuthally and radially olarized beams has been examined in [10,49]; however, its effect on the magnetic field evolution of the tightly focused APBs should be recisely examined in a future work. Although in this aer a silicon nanoshere is roosed as an illustrative examle of magnetic nanorobe, other kinds of the magnetic nanorobes (such as circular clusters made of lasmonic nanoarticles of different geometries [41]) may be advantageous in terms of increasing the magnetic near-field enhancement level, increasing the magnetic-dominant region size, or facilitating exerimental setus. 6. CONCLUSION We have characterized the focusing of an azimuthally E-olarized vector beam (APB) through a lens with secial attention on its magnetic-dominated region. When focusing the APB, the longitudinal magnetic field strength grows relatively more than the azimuthal electric field strength, leading to a region of a boosted longitudinal magnetic field. We have also elaborated on selfstanding converging APBs using PWS calculations and shown that the longitudinal magnetic field intensity sot with FWHM of 0.56λ and annular-shaed electric field intensity sot with comlementary FWHM of 0.43λ can be achieved using a converging APB with w 0 0.5λ. However, the resolution of the total magnetic field intensity at a diffraction-limited APB focus is limited by the resence of the radial magnetic field in an annular-shaed region around the beam axis with comarable magnitude to the longitudinal one. In order to enhance the longitudinal magnetic field and obtain a very high total magnetic field resolution, we have roosed to utilize a magnetically olarizable (at otical frequency) article leading to shar magnetic near-field features. Full-wave simulation results reorted here demonstrate that by lacing a subwavelength-size dense Mie scatterer (here, a silicon nanoshere) at the minimum-waist lane of a converging self-standing APB, one achieves an extremely high-resolution magnetic-dominant region with a magnetic field enhancement of about 2.3 (with resect to the incident magnetic field) and a magnetic field sot area of 0.04λ 2 at a transverse lane 0.12λ away from the scatterer surface. Such a suer-tight magnetic-dominant region, with enhanced magnetic and negligible electric near-fields, is essential for unambiguous excitation of magnetic diolar transitions in materials. This may be beneficial by adding an extra feature, based on magnetic near-field signature, to future magnetismbased scanning robe microscoy and sectroscoy systems. APPENDIX A: FIELD AT THE FOCAL PLANE OF A LENS UPON APB ILLUMINATION Let us assume that an incident araxial APB as in Eqs. (1) and (2) with beam arameter w 0;i illuminates an infinitely thin converging lens (Fig. 4). We assume the lens to be ositioned at z 0 in the transverse lane where the incident APB has its minimum CFWHM. In other words, the lens is located at the incident beam s araxial minimum-waist lane. Accordingly, the following conclusions would be still aroximately valid if the incident beam s minimum-waist lane occurs at jzj z R, leading to ζ 1 and w w 0 in Eq. (2). The electric field at the lens s araxial focal lane z f, on the right side of the lens in Fig. 4 is subsequently calculated using the Fresnel diffraction integral that in cylindrical coordinate system is written as [45] E f ρ; ϕ;z f kρ 2 k i Z i2πf e 2f Z 2π e ikf P ρ i ρ 0 ; ϕ 0 ;z 0 0 e iφ ρ 0 e 2f i kρ02 1 A e i k f ρρ 0 cos ϕ ϕ 0 ρ 0 dρ 0 dϕ 0 ; (A1) where f is the lens araxial focal distance, E i ρ 0 ; ϕ 0 ;z 0 0 is the incident APB s electric field vector at the lens lane [given by Eq. (1) with z 0 0], P ρ 0 is the uil function to account for the hysical extent of the lens, and Φ ρ 0 is the lens-induced sherical hase given in Eq. (14) required to focus the beam. Under araxial aroximation, the hase term Φ ρ 0 in Eq. (14) is aroximated as Φ ρ 0 kρ02 2f : (A2) Substituting Eq. (A2) into Eq. (A1), the Fresnel integral for the electric field at the lens s araxial focal lane can be subsequently aroximated as

11 Research Article Vol. 33, No. 11 / November 2016 / Journal of the Otical Society of America B 2275 E f ρ; ϕ;z f k i2πf ei Z Z 2π 0 0 kρ 2 2f e ikf P ρ 0 E i ρ 0 ; ϕ 0 ;z 0 0 e i k f ρρ 0 cos ϕ ϕ 0 ρ 0 dρ 0 dϕ 0 : (A3) For simlicity, let us first assume that the hysical diameter of the lens (i.e., 2a) is sufficiently larger than the beam waist of the incident beam, imlying that almost all the incident beam ower illuminates the lens. Under such assumtion, the uil function in Eq. (A3) is set to one. The incident APB s electric field E i ρ 0 ; ϕ 0 ;z 0 0 in Eq. (A3) is a suerosition of four linearly olarized LG beams, two x-olarized LG beams with l; 1; 0 and two y-olarized LG beams with l; 1; 0 [see Eq. (1)], where l and are the azimuthal and radial LG beam s mode numbers, resectively. Therefore, for the sake of simlicity, we first show the stes for a general linearly olarized LG beam with mode number l; and beam arameter w 0;i as the incident beam. An analogous treatment is readily alied to all four linearly olarized LG beams that form the APB in Eq. (1). For an incident x-olarized LG beam, the electric field at the lens lane (z 0) is given as E i ρ 0 ; ϕ 0 ;z 0 0 u l; ρ 0 ; ϕ 0 ;z 0 0 ˆx. Here, we use the following integral identities [47]: Z 2π e ilϕ0 ik e f ρρ 0 cos ϕ ϕ 0 k dϕ 0 2πi l e ilϕ J l f ρρ0 ; 0 Z jlj ρ e βρ02 L jlj kρ αρ 02 0 ρ kρ 0 ρ J l f f 12 dρ 0 kρ sgn l jlj 2 jlj 1 β jlj 1 β α jlj 2 1 f e k2 ρ 2 4βf L 2 jlj αk 2 ρ 2 4βf 2 ; (A4) α β where J l is the Bessel function of the first kind and order of l, sgn l denotes the sign of l,andl jlj is the associated Laguerre olynomial [1,12]. Accordingly, the x-olarized LG beam s field at the araxial focal lane of the lens is then written as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E f ρ; ϕ;z f V sgn l jlj 1 2! e ikρ2 2f π jlj! ffiffiffi ikf i l 1 ρ 2 jlj e e ρ 2 ffiffiffi w 0;f L jlj ρ 2 2 e ilϕ ˆx; (A5) w 0;f w 0;f w 0;f where w 0;f λ f : (A6) π w 0;i For ractical cases (as for the lens hysical arameters rovided in this aer), almost all the focused beam ower on the lens focal lane is confined in a circular region whose area is much smaller than πf λ, i.e., the focused area is within a radial distance ρ ffiffiffiffiffi λf. Therefore, the hase factor ex ikρ 2 2f in Eq. (A5) can be neglected and Eq. (A5) would clearly reresent a araxial LG beam at its minimumwaist lane with beam arameter of w 0;f. In other words, under araxial assumtion, focusing an incident LG beam through a simle lens laced at its minimum-waist lane results in another LG beam whose minimum-waist lane coincides with the lens s focal lane (z f ) and its beam arameter relates to the focal distance of the lens and the incident beam arameter through the Eq. (A6). Note that, in rincile and according to Eq. (A6), if the radial sread of the incident LG beam determined through w 0;i is in comarable length to the focus distance f, then the beam arameter of the converged beam w 0;f would be subwavelength. However, as shown in Section 4, hysical limitations accounted by using the PWS (see [12] for details on the PWS calculations) shows that there is a limit in the minimum achievable w 0;f. The electric field at the lens araxial focal lane due to an incident APB is subsequently obtained by the suerosition of the focal lane fields of its four constitutive linearly olarized LG beam terms leading to E f V ffiffiffi 2ρ π w 2 e ρ 2 w 0;f e ikf e ikρ2 2f ˆφ: (A7) 0;f In order to take into account for the hysical extent of the lens, we also consider the following uil function: 1 ρ P ρ 0 0 a; 0 ρ 0 (A8) >a; where a is the lens radius. The uil function in Eq. (A8) is exanded into a summation of Gaussian functions that come in handy in taking the Fresnel integral in Eq. (A3) analytically. Such a uil function is aroximated with a finite summation of basis Gaussian functions as [50] P ρ 0 XN n 1 A n e B n a 2 ρ 02 ; (A9) where comlex coefficients A n and B n are, resectively, exansion and Gaussian coefficients. It is demonstrated in [50] that for N 10, the uil function in Eq. (A8) is well reresented by Eq. (A9) with roer coefficients given in [50]. By substituting Eq. (A9) into Eq. (A3), the focusing field at the focal lane z f of the finite-size lens uon illumination by an incident x-olarized LG beam with l; is calculated as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E f V sgn l jlj 2! e ikρ2 2f e ikf i l 1 ffiffi ρ 2 jlj e ilϕ π jlj! w 0;f w 0;f ffiffi XN ρ 2 2 A n 1 β n jlj 1 β n 1 e ρ w 0;f 2 1 βn L jlj w 0;f 1 β 2 ˆx; n 1 n (A10) where β n B n w 2 0 a2, w 0;f is given in Eq. (A6), and the summation over the Gaussian exansion index n aears in the focal field distribution term. In this way, the araxial aroximation of the focusing field at the lens focal lane due to an incident x-olarized LG beam is conveniently exressed in series terms of Eq. (A10) for the case of a uil function of finite extent. The electric field at the lens araxial focal lane due to an incident APB illumination is then calculated by the suerosition of the four constitutive linearly olarized LG beam terms.

12 2276 Vol. 33, No. 11 / November 2016 / Journal of the Otical Society of America B Research Article Fig. 16. Normalized magnitude of the transverse field sectrum Ẽ for APBs with (a) w 0 3λ, (b) w 0 0.9λ, and (c) w 0 0.5λ. Note that these APBs are made mainly by the roagating sectrum (such that k 2 x k 2 y <k 2 0 ), and therefore the sectral magnitude rofiles are basically similar at any transverse lane (here, we show only the roagating sectrum region). Fig. 17. Ratio of the APB s sectral energy er unit length (in the z direction) confined in the roagating sectrum to its total sectral energy er unit length (so-called figure of APB s sectral energy) defined in Eq. (B5) as a function of the beam arameter w 0. APPENDIX B: SPECTRAL INTERPRETATION OF THE BEAM PROPAGATION IN THE NONPARAXIAL REGIME The electric field distribution for APB given in Eq. (1) reresents a self-standing beam in the araxial regime. Therefore, it is imortant to address limitations of these araxial exressions in the cases of beams with very tight satial extents (small w 0 ). With this goal in mind, we reort in Fig. 16 the normalized magnitude of the lane-wave sectrum for APBs, i.e., the 2D Fourier transform of the transverse field of the APB as Ẽ k x ;k y ;z Z Z E x;y;z e ik xx ik y y dxdy (B1) (see [12] for details on the numerical calculation of the integral). In Fig. 16, we show the wavenumber sectrum of three APBs with different beam arameters: (a) w 0 3λ, (b) w 0 0.9λ, and (c) w 0 0.5λ. It is observed from Fig. 16 that the lane-wave sectrum of the tighter beam (beam with smaller w 0 ) covers a wider region in the k x k y lane, where k 0 2π λ is the free-sace wavenumber. Moreover, the field sectral distribution for all three beams is well confined in the roagating wave sectrum with k 2 x k 2 y <k 2 0. Hence, they are mainly constructed by roagating sectral comonents only. They roagate along the z axis with ex ik z z, where k z is real and evaluated as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k z k 2 0 k2 x k 2 y : (B2) All sectral magnitude distributions in Fig. 16 are reresentative at any z lane as these field sectral distributions roagate with no magnitude variation (imlied by the roagator with magnitude j ex ik z z j 1). Let us now consider these three field distributions and look at the araxial wave aroximation. The araxial wave equation is valid under the assumtion that most of the field sectrum is confined to a region with k 2 x k 2 y k 2 0. Under this condition, the accurate PWS evaluation can be aroximated with the araxial field exression of a roagating beam as in Eqs. (1) and (2) [46]. Indeed, the required condition for deriving the araxial field exressions using PWS calculations is to aroximate Eq. (B2) as k z k 0 k 2 x k 2 y 2k 0 : (B3) It is observed from Fig. 16 that for w 0 0.9λ and w 0 0.5λ cases, the sectral distributions cannot be fully confined to a region with k 2 x k 2 y k 2 0 in contrast to the case with w 0 3λ. Therefore, the rediction of the araxial beam roagation is exected to deviate from the actual roagation of the beam, much more for w 0 0.5λ than for w 0 0.9λ and much more for w 0 0.9λ than for w 0 3λ. The sectra with w 0 0.5λ and w 0 0.9λ generate tight field sots, but the z location of the tight sots cannot be accurately redicted by the araxial field equations. In articular, when considering a converging beam with w 0 0.5λ or 0.9λ, we can exect that the actual tight sot location (minimum-waist lane) will be formed closer to the reference lane than the one redicted by the araxial exressions. This is due to the fact that the field of the APB with w 0 0.5λ or 0.9λ constitutes lane-wave sectral comonents with large transverse wavenumbers that are not modeled accurately in the araxial field exressions and roagate at larger incidence angles with resect to the beam axis (the z axis) and thus a tight sot forms closer than the one redicted by araxial field exressions. The time-average sectral energy of the APB er unit length along the z direction is calculated as W 1 4 ε 0jẼj μ 0j Hj 2 : (B4) We define here the figure of APB s sectral energy as the ratio of the APB s sectral energy er unit length in the roagating sectrum to its total sectral energy er unit length: RR F W k 2 x k 2 y <k 2 0 R R Wdk x dk y Wdk : x dk y (B5) Figure 17 shows the figure of APB s sectral energy as a function of the beam arameter. It is observed that for w 0 0.5λ, more than 95% of the APB s sectral energy is confined in the roagating sectrum. Funding. W. M. Keck Foundation (USA). Acknowledgment. The authors would like to thank Comuter Simulation Technology (CST) of America, Inc. for roviding CST Microwave Studio, which was instrumental in this work.