Focused azimuthally polarized vector beamand spatial magnetic resolution belowthe diffraction limitMEHDI VEYSI, CANER GUCLU, AND FILIPPO CAPOLINO*Department of Electrical Engineering and Computer Science, University of California, Irvine, California 92697, USA*Corresponding author: f.capolino@uci.edu

Received 10 May 2016; revised 31 August 2016; accepted 7 September 2016; posted 15 September 2016 (Doc. ID 264935);published 11 October 2016

An azimuthally electric-polarized vector beam (APB), with a polarization vortex, has a salient feature that it con-tains a magnetic-dominant region within which the electric field is ideally null while the longitudinal magneticfield is maximum. Fresnel diffraction theory and plane-wave spectral calculations are applied to quantify fieldfeatures of such a beam upon focusing through a lens. The diffraction-limited full width at half-maximum(FWHM) of the beam’s longitudinal magnetic field intensity profile and complementary FWHM of the beam’sannular-shaped total electric field intensity profile are examined at the lens’s focal plane as a function of the lens’sparaxial focal distance. Then, we place a subwavelength dense dielectric Mie scatterer in the minimum-waistplane of a self-standing converging APB and demonstrate for the first time, to the best of our knowledge, thata very-high-resolution magnetic near-field at optical frequency is achieved with total magnetic near-field FWHMof 0.23λ (i.e., magnetic near-field spot area of 0.04λ2) within a magnetic-dominant region located one radius(0.12λ) away from the scatterer. In particular, the utilization of the nanosphere as a magnetic nanoantenna(so-called magnetic nanoprobe) illuminated by a tightly focused APB is instrumental in boosting the photoin-duced magnetic response and suppressing the electric response of a sample matter. The access to the weakphotoinduced magnetic response in sample matter would add extra degrees of freedom to future optical photo-induced force microscopy and spectroscopy systems based on the excitation of photoinduced magnetic dipolartransitions. © 2016 Optical Society of America

OCIS codes: (140.3295) Laser beam characterization; (260.2110) Electromagnetic optics; (070.2580) Paraxial wave optics;

(080.3630) Lenses; (080.4865) Optical vortices; (180.4243) Near-field microscopy.

http://dx.doi.org/10.1364/JOSAB.33.002265

1. INTRODUCTION

Vector beams [1–7] are a class of optical beams whose polari-zation profiles on the transverse plane, perpendicular to thebeam axis, can be engineered to have an inhomogeneousdistribution. Among them, beams with cylindrical symmetry(so-called cylindrical vector beams), particularly radially[3,4,8–10] and azimuthally [11–13] electric-polarized vectorbeams, are exceptionally important in the optics community.Owing to the presence of the longitudinal electric field com-ponent, a radially polarized vector beam with a ring-shapedfield profile after tight focusing through a lens provides a tighterelectric field spot compared to the well-known linearly and cir-cularly polarized beams [8,9]. Such a beam has been extensivelyexamined under tight focusing and has found many prominentapplications in particle manipulation, high-resolution micros-copy, and spectroscopy systems [3,5,6,8,9,14–24]. Here, we

are particularly interested in studying the azimuthally electric-polarized vector beam primarily due to its unique magneticfield features: a strong longitudinal magnetic field where theelectric field is null. In the following, we denominate such abeam simply as an azimuthally polarized beam (APB) referringto the local orientation of its electric field vector. As schemati-cally shown in Fig. 1, APBs possess an electric field purelytransverse to the beam axis and a strong longitudinal magneticfield component in the vicinity of the beam axis where the trans-verse electric and magnetic fields are negligible and even vanishon the beam axis [12]. This so-called magnetic-dominant regionis characterized by the presence of a tight magnetic field withlongitudinal polarization. Focusing an APB through a lens boostsits longitudinal magnetic field component relatively more thanits transverse electric and magnetic fields [12].

Due to such a unique property, the APB may be beneficialby adding an extra feature to future spectroscopy and scanning

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probe microscopy systems [25,26] based on the excitation ofmagnetic dipolar transitions [11,12,27,28]. At optical frequen-cies, photoinduced magnetic dipolar transitions in matter areweaker than their electric counterparts [28–30] and there-fore require an excitation beam with an enhanced magnetic-dominant region to be explicitly excited [28]. In this regard,APBs are a very suitable choice for the illumination beam insuch spectroscopy and scanning probe microscopy systems.Even though various methods have been proposed to generateAPBs [12,31–40], characterization of the magnetic field ofthese beams under tight focusing, to best of the authors’ knowl-edge, remains to be fully elucidated. This study is the basis forthe successful implementation of magnetically sensitive nanop-robes at optical frequency, which are crucial in the developmentof magnetism-based spectroscopy applications and the study ofweak photoinduced magnetism in matter [27,41,42].

In this paper, we report the diffraction-limited tight field(especially magnetic field) features of an APB, represented interms of paraxial Laguerre–Gaussian (LG) beams, with beamparameter w0 that is a measure of the spatial extent of the beamin the transverse plane at its minimum waist. Two figures ofmerit are used in this paper to quantify the field featureson transverse planes: (i) the full width at half-maximum(FWHM) of the longitudinal magnetic field intensity and(ii) the complementary FWHM (CFWHM) of the annular-shaped total electric field intensity. Keeping in mind that fora very small beam parameter w0 the expressions obtained viaparaxial approximation may not be accurate, we also report re-sults using the accurate analytical-numerical plane-wave spec-tral (PWS) calculation [43], which is analogous to the Richardsand Wolf theory [44].

We first elaborate on the diffraction-limited tight focus of anAPB through a converging lens using both paraxial Fresnel dif-fraction integral formulation, leading to analytical assessments,and the accurate PWS calculations (see [12] for more details onPWS). We demonstrate using the Fresnel integral under para-xial approximation that upon focusing through a lens an inci-dent APB converts to another APB whose beam parameter islinearly proportional to the lens paraxial focal distance andinversely proportional to the incident APB parameter (seeAppendix A). The minimum-waist plane position of the focusedbeam predicted by the Fresnel integral coincides with the lensparaxial focal plane, which deviates from the actual focal planeposition calculated by PWS. The figures of merit of an APBfocused by a lens are therefore calculated both by the Fresnelintegral at the lens’s paraxial focus and by the PWS at the actualfocal plane as a function of the lens paraxial focal distance.

In addition to the case of focusing an APB by a lens men-tioned above, the tight field features of a self-standing converg-ing APB, as schematically shown in Fig. 1, are also examined,and its figures of merit are calculated using the paraxial LGbeam expressions and the PWS calculations at the mini-mum-waist planes predicted by the respective methods.Recently, it has been experimentally confirmed that cylindricalvector beams may selectively excite the electric or magneticdipolar resonances of a subwavelength-sized dense dielectricnanosphere (e.g., a silicon nanosphere) [24]. In this paper, weuse a silicon nanosphere as a magnetic nanoantenna (so-called

magnetic nanoprobe) and place it at the focus of a convergingAPB that selectively excites a magnetic dipolar resonance in thenanosphere as in [24]. The aim is to achieve a subwavelengthmagnetic field resolution. In general, such a subwavelength-sized scatterer hosts a magnetic Mie resonance with a circu-lating electric displacement current in addition to an electricdipolar resonance. However, the latter is not excited by anAPB due to its cylindrical symmetry, which ideally leads to anull average displacement current over the nanosphere. The in-duced electric displacement currents with a net magnetic dipolemoment in the Si nanosphere along the z direction are shownto boost not only the total longitudinal magnetic field but alsothe spatial magnetic field resolution below the diffraction limitin the vicinity of the scatterer. A total magnetic field enhance-ment of about 2.3 (with respect to the total incident magneticfield) and a total magnetic field spot area as small as 0.04 λ2 areachieved within a magnetic-dominant region, evaluated at atransverse plane one nanosphere radius (0.12 λ) away from thescatter surface.

Note that FWHM is an effective feature to characterize themagnetic near-field intensity. The FHWM is also here used as ashorthand measure of resolution, i.e., the minimum resolvabledistance between two closely spaced point sources because theside-lobe peak of the magnetic near-field intensity profile is forall cases by far less than half of its main peak. Throughout thepaper, we also consider time harmonic fields with an exp�iωt�time dependence, which is suppressed for convenience.Furthermore, bold symbols denote vectors and hats (^) indicateunit vectors.

2. CHARACTERIZATION OF AN APB

APB is here expressed as a superposition of a left- and a right-hand circularly polarized beam, carrying orbital angularmomentum (OAM) with orders of �1 and −1, respectively.In paraxial regimes, OAM-carrying beams are analyticallyrepresented as LG beams [1]. Thus, the APB’s electric fieldis expressed in terms of self-standing paraxial LG beams incylindrical coordinate system as [12]

E � −iffiffiffi2

p

2�u−1;0eRH − u1;0eLH�eikz ; (1)

where eRH � �x� iy�∕ ffiffiffi2

pand eLH � �x − iy�∕ ffiffiffi

2p

are,respectively, right- and left-hand circularly polarized unitvectors, and the LG beam expression is

Fig. 1. Schematic of a converging azimuthally electric-polarized vec-tor beam (APB), with a longitudinal magnetic field on its axis.

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u�1;p�0 �Vffiffiffiπ

p 2ρ

w2 e−�ρ∕w�2ζe−2i tan−1�z∕zR�e�iφ;

w � w0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� �z∕zR�2

q; ζ � �1 − iz∕zR�; (2)

where V is an amplitude coefficient (in volts), zR � πw20∕λ is

the Rayleigh range, and k � 2π∕λ and λ are the wavenumberand wavelength in the host medium, respectively. The beamparameter w0 controls the transverse spatial extent of the beamat its minimum-waist plane. Vaguely speaking, w0 correspondsto the minimum waist, which is very well defined for the fun-damental Gaussian beam (FGB). Since the actual waist of theAPB differs from w0, we prefer to call it simply as “beamparameter” because this difference is of relevance in this paper.Here, the term “beam waist” is reserved for the minimum of theactual waist size as discussed next.

The electric field in Eq. (1) is equivalently expressed as [12]

E � Eφφ � Vffiffiffiπ

p 2ρ

w2 e−�ρ∕w�2ζe−2i tan−1�z∕zR�eikzφ; (3)

which clearly shows the purely azimuthal polarization of thebeam. The electric field intensity profile of an APB is plottedat the beam’s minimum-waist plane (i.e., z � 0) in Fig. 2(a).It is observed that the APB’s electric field has an annular-shapedintensity profile whose CFWHM is of interest to us as a mea-sure of the beam’s tightness. The APB examined in Fig. 2 iscarrying a power of 1 mW, which is obtained by setting V �0.89 V in Eq. (3) and its beam parameter is set to w0 � 0.9 λ.In Appendix B and Section 4 of this paper, we show that theconverging APB expressed by Eq. (1) with such an illustrativebeam parameter (w0 � 0.9 λ) represents, by a good approxima-tion, a self-standing beam. The strength of the APB’s electricfield given in Eq. (3) is proportional to ρ exp�−ρ2∕w2� in anygiven z transverse plane, and it reaches its maximum

jEφ�ρM ; z�j � jV jffiffiffiπ

p 2ρMw2 e−�ρM∕w�2 �

ffiffiffi2

pffiffiffiffiffiπe

p jV jw

(4)

at ρM � w∕ffiffiffi2

p. Therefore, on the minimum-waist plane (i.e.,

z � 0) the electric field magnitude peaks at ρM � w0∕ffiffiffi2

p,

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 incylindrical coordinates, yielding a longitudinal magnetic fieldcomponent as [12]

Hz �−Vffiffiffiπ

p 4iw2ωμ

�1 −

�ρ

w

�2

ζ

�e−�ρw�2ζe−2i tan−1

�zzR

�eikz (5)

alongside a radial magnetic field component as

H ρ � −1

ηEφ

�1� 1

kzR

ρ2 − 2w20

w2

�: (6)

It is observed from Eq. (6) that for kzR ≫ 1 the radialmagnetic field component follows the electric field profile ofthe beam. In summary, the APB possesses only Eφ, Hz , andH ρ field components. The intensity of the APB’s longitudinalmagnetic field [given in Eq. (5)] is plotted in Fig. 2(b), where itpeaks on the beam axis (ρ � 0) and is characterized by itsFWHM. The maximum of the longitudinal magnetic fieldstrength at any z is given by

jHz�ρ � 0; z�j � 4jV j∕�w2ωμffiffiffiπ

p � (7)

and is thus inversely proportional to w2. It is observed fromEq. (7) that the longitudinal magnetic field of the APB peaksat the beam’s minimum-waist plane (i.e., z � 0, wherew � w0), where its magnitude is inversely proportional tothe square of the beam parameter w2

0. The transverse magneticfield [which is purely radial and given in Eq. (6)] increasestogether with the electric field (which is purely azimuthal)as the radial distance ρ from the beam axis increases and peaksaway from the beam axis alongside the azimuthal electric field,as shown in Fig. 2(c). By duality, this is analogous to the case ofthe radially polarized beam in which electric field intensity ispurely longitudinal on the beam axis and its transverse compo-nent peaks off the beam axis [3,4,6,11]. Here, we define theCFWHM for the annular-shaped electric field intensity profileof the APB as the width across its null, where the field intensityrises to the half of its maximum, i.e., to 0.5jEφ�ρM ; z�j2 [seeEq. (4) and Fig. 2(a)]. In addition, the FWHM of the longi-tudinal magnetic field intensity is also calculated as the widthacross its peak on the beam axis, where the longitudinal mag-netic field intensity drops to the half of its maximum, i.e., to0.5jHz�ρ � 0; z�j2 [see Fig. 2(b)]. Based on the azimuthallypolarized electric field and the longitudinally polarized mag-netic field expressions given, respectively, in Eqs. (3) and (5),the CFWHM of the electric field intensity and the FWHM ofthe longitudinal magnetic field intensity at the minimum-waistplane are calculated and given by

CFWHM�Eφ�jz�0 ≈ 0.68w0;

FWHM�Hz�jz�0 ≈ 0.81w0: (8)

One may be also interested in the ratio of the longitudinalmagnetic field on the beam axis, where it is maximum, to themaximum of the electric field at ρ � ρM . This ratio, normal-ized with respect to the inverse of the host-medium waveimpedance η−1 �

ffiffiffiffiffiffiffiffiε∕μ

p, is equal to

ηjHz�ρ � 0; z�jjEφ�ρM ; z�j �

ffiffiffi2

p

π

λ

we1∕2 ≈ 0.74

λ

w: (9)

Note that such ratio is inversely proportional to w, and itreaches its maximum at z � 0, i.e., in the minimum-waistplane. Therefore, the maximum magnitude of the longitudinalmagnetic field increases relatively more than the maximum

Fig. 2. Intensity profile of (a) the electric field, (b) the axis-confinedlongitudinal magnetic field, and (c) the purely radial transverse mag-netic field for an APB carrying 1 mW power and with beam parameterof w0 � 0.9 λ at λ � 523 nm.

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magnitude of the electric field as w0 decreases (tighter beams).Note that decreasing w0 also has the effect of decreasing thearea of the longitudinal magnetic field spot.

Finally, we should note that on the minimum-waist plane(z � 0) the ratio ηjHz�ρ; 0�j∕jEφ�ρ; 0�j is equal to unity atthe radial distance

ρ � w0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�

�πw0

2λ

�2

s−πw0

2λ

!(10)

and inside this radius, the longitudinal magnetic to total electricfield contrast ratio for APB is larger than the magnetic to elec-tric field contrast ratio (the admittance) of a plane wave 1∕η.The optical power carried by the APB is calculated by theintegral of its longitudinal Poynting vector over its mini-mum-waist plane as

P � 1

2

Z2π

0

Z∞

0

Ref−Eφ�H ρ��gz�0ρdρdφ: (11)

After substituting the APB’s azimuthal electric and radialmagnetic fields formulas given in Eqs. (3) and (6) intoEq. (11), the power carried by the APB is evaluated as

P � jV j22η

1 −

1 − 12

R2π0

R∞0

jEφjw0jV j

2ρ3dρdφ

�πw0∕λ�2!: (12)

After a change of variable from 2�ρ∕w0�2 to t , the integralterm in Eq. (12) is found to be equal to 0.5Γ�3� � 1, whereΓ�·� is the gamma function. Therefore, Eq. (12) is reduced to

P � jV j22η

�1 −

1

2�πw0∕λ�2�: (13)

Equation (13) clearly shows that power carried by the APB isexplicitly expressed as a function of w0∕λ and the absolute valueof the amplitude coefficient jV j. Therefore, the APB’s ampli-tude coefficient V is obtained for certain beam parameter w0

and required power using Eq. (13).In order to have a better assessment of the APB’s significance

in providing a magnetic-dominant region, here we compare anAPB with a traditional FGB of equal powers and beam param-eters. The electric and magnetic field distributions of a paraxialAPB and a FGB at their paraxial minimum-waist planes, i.e., atz � 0, are compared in Fig. 3 for two illustrative beam param-eters set to (top row) w0 � 0.9λ and (bottom row) w0 � 0.5λ.Here, the APB and the FGB carry equal powers of 1 mW[see Eq. (13)].

In order to have azimuthally symmetric magnetic field dis-tribution for the FGB, we consider circularly polarized FGB inFig. 3; however, similar conclusions would be obtained if weused linearly polarized FGB. In contrast to the FGB whosemagnetic and electric fields peak on the beam axis (i.e., thez axis), the APB contains a pure longitudinal magnetic fieldcomponent on the beam axis where its electric field vanishes.The magnetic-to-electric field intensity ratio normalized to thatof a plane wave is also plotted for the APB and the FGB inFig. 3 (third column) varying radial distance from the beamaxis. Note that the magnetic-to-electric field intensity ratio ofthe FGB is very close to that of a plane wave. In contrast, theAPB has a very large magnetic-to-electric field intensity ratio in

the vicinity of the beam axis denoting the magnetic-dominantregion. This ratio for the APB tends to infinity when ρ → 0.For w0 � 0.9λ [Fig. 3 (top row)], even though the strength ofthe APB’s magnetic field on the beam axis is half of that of theFGB carrying the same power, the APB uniquely has only mag-netic field and no electric field there, which is an importantfeature that can be used in various applications. In addition,it is observed from Fig. 3 (top row) that the FWHM of thetotal magnetic field for the APB with w0 � 0.9λ is larger thanthat for the FGB with the same w0; this is attributed to the factthat the APB contains an annular-shaped radial magnetic fieldcomponent [see Fig. 2(c)]. However, decreasing the beamparameter (tightening the beam) from w0 � 0.9λ to 0.5λboosts the longitudinal magnetic field component relativelymore than the radial one and therefore significantly decreasesthe FWHM of the total magnetic field, as shown in Fig. 3 (bot-tom row). To further reduce the FWHM of the total magneticfield of the APB approaching that of its longitudinal magneticfield component, one approach might be to use ring-shapedlenses with high numerical apertures for focusing of the APB.This technique has been used for generating very sharp electricfield focuses using radially polarized beams [8,9].

3. FOCUSING AN APB THROUGH A LENS

In this section, we aim at characterizing magnetic and electricfields of an APB at the focal plane of a lens through whichthe APB is focused. To have first an analytical assessment,we use the Fresnel integral as summarized in Appendix Aand calculate the fields at the lens’s paraxial focal plane. Weshow in Appendix A that a lens, under paraxial approximation,converts an incident APB, whose minimum-waist plane occursat the lens surface, to another converging self-standing APBwhose paraxial minimum-waist plane coincides with the lens’s

Fig. 3. Comparison between a FGB and an APB of equal powers(1 mW) at λ � 523 nm and beam parameters at their minimum-waistplanes (i.e., z � 0): (top row) w0 � 0.9λ and (bottom row)w0 � 0.5λ. Strength of the total electric field (first column), strengthof the total magnetic field (second column), and the ratio of the totalmagnetic to the total electric field intensities normalized to that of aplane wave (third column). (Note how this ratio grows for the APBwhen approaching the beam axis.)

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paraxial focal plane. Results are schematically represented inFig. 4 for a specific example where we show the total electricand the longitudinal magnetic field magnitudes of the APBbefore and after focusing through the lens.

Next, in order to confirm the analytical calculations andprovide a guide to where the Fresnel integral expressions areaccurate, we characterize the APB upon focusing through aconverging lens using accurate PWS calculations. As for thePWS calculations, we assume the thin lens approximation suchthat each ray entering one side of the lens exits the other side atthe same transverse (ρ, φ) coordinates as the entrance position.We model the transmission through the lens by imposing aphase shift, which varies in the radial direction, added tothe ρ-dependent phase of the incident APB. The transmissionphase shift that is added, relative to a spherically convergingwave, is given by

Φ�ρ� � −2πfλ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� ρ2

f 2

s− 1

!; (14)

where f is the paraxial focal distance of the lens (on the rightpanel of Fig. 4), ρ is the local radial coordinate of the lens, and λis the wavelength in the host medium on the right side of thelens. Note that we are assuming that the lens does not varythe ρ-dependent amplitude of the incident APB’s field acrossthe lens. We also must remember that in the Fresnel integralequation, the phase term in Eq. (14) is paraxially approximatedas a quadratic phase [see Eq. (A2)] term [45]. The Fourier andinverse Fourier transform integrals in PWS calculations (seeEqs. (24) and (25) in [12]) are then numerically calculated viaa two-dimensional FFT algorithm, where the spatial domainsize and the spatial resolution are 102.4λ × 102.4λ and λ∕20,

respectively. The corresponding spectral domain size and thespectral resolution are 20k × 20k and ∼0.01k, respectively, ex-tending to evanescent spectral components as big as 10k.Moreover, to model the hard-edged aperture, the electric fieldis assumed null outside of the overall lens aperture in thelens plane.

We now characterize the FWHM of jHz j2 and theCFWHM of jEj2 at the lens focal plane for an incidentAPB. As a representative example, we set the lens radius a equalto 40λ and characterize the focusing beam at the lens’s focalplane as the lens’s paraxial focal distance f changes. The inci-dent APB has a beam parameter of w0;i � 29λ such that thebeam cross section is much wider than the wavelength and90% of the incident beam power illuminates the lens surface.In Fig. 5, we plot the FWHM of jHz j2 and CFWHM of jEj2calculated at the lens’s focal plane as a function of the lens radiusto focal distance ratio a∕f , where a is kept constant and f isvaried. We recall that the right side of Fig. 4 corresponds to thefield maps for a specific case with a � f ∕2, which is a point onthe curves reported in Fig. 5. The quantities plotted in Fig. 5are calculated using both the Fresnel integral formula [given inEq. (A10)] at the lens’s paraxial focal plane (z � f ) and PWScalculations (refer to [12] for more details on PWS) at the lens’sactual focal plane.

It is observed from Fig. 5 that the Fresnel integral results(denoted by FI) agree very well with the accurate PWS results,especially for large focal distances (small a∕f ). We also observefrom the PWS results that for the case with f � a, the FWHMof the longitudinal magnetic field intensity and CFWHM ofthe total electric field intensity at the lens actual focal planeare 0.715λ and 0.53λ, respectively. Note that the actual focalplane, obtained from PWS calculations, is slightly displacedfrom the lens’s paraxial focal plane as described in the next sec-tion. In Appendix B, we elaborate more on this as we examinethe plane-wave spectrum of converging beams.

4. SELF-STANDING CONVERGING APB

As we discussed in the previous section, a lens transforms anincident APB to another converging self-standing APB (seeFig. 4 and Appendix A). Such transformation simplifies thecalculations of the focusing beam due to a lens, assuming

Fig. 4. Schematic of a converging lens transforming an incidentAPB with beam parameter w0;i into another converging APB withbeam parameter w0;f . The magnitudes of the total electric (whichis purely azimuthal) and the longitudinal magnetic fields are plotted.The radial component of the magnetic field, also experiencing focus-ing, is not shown here for brevity. In this representative example, theincident APB carries 1 mW power, and the lens radius and focal dis-tance are set at a � 40λ and f � 80λ, respectively. The beam param-eters of the incident and focusing APBs are w0;i � 29λ andw0;f � 1.3λ, respectively.

Fig. 5. (a) FWHM of the longitudinal magnetic field intensityjHz j2, and (b) CFWHM of the annular-shaped electric field intensityjEj2 calculated using (i) PWS at the actual focal plane and (ii) theFresnel integral (FI) at the lens’s paraxial focal plane, upon illuminatingthe lens by an incident APB, varying the normalized lens’s focal dis-tance f .

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the paraxial approximation. Therefore, in this section, weexamine propagation of a self-standing converging APB, assum-ing it is focused by the lens, and quantify its properties at itsminimum-waist plane. We show some important tight fieldfeatures of self-standing converging APBs as a function of theirbeam parameter w0;f (as shown in Fig. 4), where we payparticular attention to the FWHM of their longitudinal mag-netic fields at the minimum waist. Since we only elaborate on aself-standing converging beam, we drop the subscript f anddenote the converging beam parameter simply as w0. Note that,in general, the evanescent components of the field transmittedthrough lens are negligible and filtered out as the beam prop-agates toward the focal plane. The spectral components of theconverging APBs [expressed by Eq. (1)] are examined inAppendix B, where we show that more than 95% of the spectralenergy of the converging APBs with w0 ≥ 0.5λ is confined inthe propagating spectrum. Thus, in the subsequent studies, thebeam parameter of the converging APB is set larger or equal to0.5λ. The results pertaining to the paraxial beam propagationare also compared to those obtained from the analytical-numerical computation based on the PWS. We assume toknow the initial APB’s field distribution, with converging fea-tures, at a certain z plane (the so-called reference plane) andobserve the beam propagating toward its minimum-waist planein �z direction. In other words, we investigate the convergingproperties of the beam on the right side of the lens in Fig. 4.

We first assess the validity of the paraxial approximation forAPBs as in Eq. (1). It is known that the paraxial approximationfor a beam holds under the following condition [2,46,47]:����2k ∂ψ∂z

���� ≫���� ∂2ψ∂z2

����; (15)

where E � ψeikz represents paraxial field distribution for abeam propagating in the �z direction. In order to determinethe validity range of the paraxial field, we define a paraxialityfigure as

F p �����2k ∂ψ∂z

����∕���� ∂2ψ∂z2

����; (16)

which is a function of local coordinates. We also define thenormalized weighted average figure of the paraxiality at eachtransverse z plane as

F p;ave �R∞−∞R∞−∞ Fpjψj2dxdyR

∞−∞R∞−∞ jψj2dxdy ; (17)

where the numerator is the average paraxiality figure weightedby the intensity of the transverse field, and the denominator isthe total weight of the transverse field intensity with respect towhich we normalize the weighted average paraxiality figure.The value of F p;ave for the paraxial APB’s electric field inEq. (1) is calculated and plotted in Fig. 6 as a function ofthe beam parameter w0 at the beam’s paraxial minimum-waistplane z � 0. The larger the paraxiality figure F p;ave is, the betterthe paraxial approximation is. It is observed from Fig. 6 thatF p;ave ≥ 50 [i.e., log10�F p;ave� ≥ 1.7] for beam parameterslarger than 0.9λ. We assume that F p;ave values larger than50 represent reasonably valid paraxial beams for practical pur-poses. Thus, for such values of w0, the paraxial electric field

expression given in Eq. (1) represents a self-standing APB’s fielddistribution with a good approximation. Remarkably, the sig-nature of this “validity range”manifests itself in the comparisonof the paraxial beam propagation and the accurate PWS resultsdiscussed in the following.

We now examine the magnetic and electric field features of aself-standing converging APB at its minimum-waist plane as afunction of the beam parameter w0 (for w0 ≥ 0.5λ). With thisin mind, we characterize self-standing converging APB usingPWS calculations. We start with an APB’s paraxial transversefield distribution on a transverse reference plane located at z �zr (zr < zf ; see Fig. 1 for zf ) given by Eq. (1). Subsequently, theevolutions of the beam’s magnetic and electric fields in the pos-itive z direction are examined using the PWS calculations. Thelocation of the actual minimum-waist plane of a converging APB“launched” from a reference plane at zr � −3.5λ with the fielddistribution given in Eq. (1) is calculated using PWS and plottedin Fig. 7 as a function of the beam parameter w0. We observethat the actual minimum-waist plane of the converging APBdoes not occur at z � 0, that is the location of the focus pre-dicted by the paraxial field expression. This difference is attrib-uted to the presence of plane-wave constituents with largetransverse wavenumbers in the field spectrum of the convergingAPB, which are not properly modeled in the paraxial fieldexpressions (see Appendix B for more details on the spectral con-tent of the APB). For an APB, with decreasing w0 a largeramount of constitutive propagating plane-wave spectral compo-nents of the beam’s field will have large transverse wavenumbers.

Fig. 6. Normalized weighted average figure of paraxiality F p;ave (inlogarithmic scale) for a converging APB at the beam’s paraxial mini-mum-waist plane (z � 0) as a function of the beam parameter w0.

Fig. 7. Actual position of the minimum-waist plane of the beam zfas a function of the beam parameter w0 for both APB and FGB (cal-culated using the PWS). The minimum-waist plane estimated by usingthe simple paraxial field expression is at z � 0.

2270 Vol. 33, No. 11 / November 2016 / Journal of the Optical Society of America B Research Article

Hence, the difference between the actual minimum-waistplane’s location (here, denoted by zf ) and the one predicatedby the paraxial field expressions (here at z � 0) becomesmore significant because of the loss of accuracy of the paraxialapproximation with decreasing w0. Such a difference is notunique to the APBs, and a similar difference is observed fortraditional FGBs, as shown in Fig. 7. We observe that the dif-ference between the actual and the paraxial minimum-waistplane’s positions for a FGB is also increasing as w0 decreases.However, the difference between the actual minimum-waistplane’s position and the paraxial one is larger for an APB thanfor a FGB with an equal w0. This is due to the fact that thetransverse wavenumber spectrum of an APB’s field is broaderthan that of a FGB’s field with the same beam parameter;hence, the paraxial approximation is coarser for the APBcompared to the FGB.

Next, the FWHM of jHz j2 and the CFWHM of jEj2 of theconverging APB are calculated using both paraxial and PWScalculations and plotted in Fig. 8. The FWHM and theCFWHM in the PWS calculations are evaluated at the actualminimum-waist plane of the APB (z � zf ), which depends onw0 (see Fig. 7). Instead, the FWHM and CFWHM under theparaxial approximation are evaluated using Eq. (1) at z � 0 forall the w0 cases. It is observed from the paraxial calculations thatthe FWHM and CFWHM curves decrease monotonically asthe beam parameter w0 decreases. However, in practice, thedecrease in FWHM of the longitudinal magnetic field intensityprofile as well as CFWHM of the electric field intensity profileis hampered by an ultimate limit imposed by the diffraction ofthe beam. It is observed from the PWS curves in Fig. 8 that theFWHM and CFWHM of the converging APB are saturated bythe diffraction to about 0.56λ and 0.43λ, respectively, despitethat the paraxial approximation estimates much smallerFWHM and CFWHM. Thus, according to accurate PWS cal-culations, a longitudinal magnetic field intensity profile withFWHM as small as 0.56λ (spot area of about 0.25λ2) is achiev-able with w0 � 0.5λ. The spot area is defined here as thecircular area whose diameter is equal to the FWHM.

Here, based on what is discussed in Appendix B, we stressthat the transverse wavenumber spectrum of the APB’s field inEq. (1) with very small w0 (w0 < 0.5λ) is not confined only inthe propagating wavenumber spectrum, and it starts to extendto the evanescent spectral region and therefore is not shown in

Fig. 8. However, the spatial field distribution in Eq. (1) with w0

as small as 0.5λ has wavenumber spectral constituents stillconfined in the propagating spectrum (see Appendix B), andit has a relatively large normalized weighted average figure ofparaxiality F p;ave ≈ 14. Therefore, though it may not representa strictly self-standing APB, the paraxial approximation is nottoo coarse.

When the beam parameter w0 is larger than 0.9λ, the para-xial curves for the FWHM and the CFWHM in Fig. 8 followthe accurate PWS ones very well, and they start to deviate fromPWS curves when w0 decreases to smaller values, which is inagreement with our finding in Fig. 6. In order to clarify theeffect of beam parameter w0 on different magnetic fieldcomponents of the APB, in Fig. 9 we plot the strength ofthe longitudinal (Hz) and the radial (H ρ) magnetic field com-ponents as well as the strength of the azimuthal electric fieldnormalized to the host-medium wave impedance for two illus-trative w0 values at λ � 523 nm, using PWS calculations. Werecall that the APB has a Hz profile that peaks on the beam axis(ρ � 0), whereas its transverse magnetic field component ispurely radial and peaks off the beam axis. It is observed fromFig. 9 that the longitudinal magnetic field spot areas as small as0.25λ2 and 0.49λ2 are obtained with converging APBs with w0

of 0.5λ and 0.9λ, respectively. However, since APB possesses aradial magnetic field component over an annular-shaped regionin addition to the longitudinal one (see Fig. 9), the FWHM ofthe total magnetic field is always larger than the FWHM of thelongitudinal magnetic field component.

It is also observed from Fig. 9 that when w0 of the APBdecreases from 0.9λ to 0.5λ, the strength of its longitudinalmagnetic field component increases by about 2.2 times, whichis relatively more than the increase in the strength of its radialmagnetic field component (1.24 times). Indeed, as the beamparameter w0 decreases, the plane-wave spectral distributionof the APB includes large transverse wavenumbers. For smallerbeam parameters, such as w0 � 0.5λ, a larger portion of theconstitutive transverse electric (TE, with respect to z) planewaves in the spectrum of the APB possess large transverse wave-numbers, meaning that they propagate in directions with largerangles α with respect to the beam axis, as shown in Fig. 10.Therefore, the magnetic fields of the TE constitutive planewaves, which are perpendicular to the plane-wave propagationdirections, are more aligned with the beam axis. The depth of

Fig. 8. PWS and paraxial calculations for (a) the FWHM of thelongitudinal magnetic field intensity and (b) the CFWHM of the an-nular-shaped electric field intensity of the converging APB as a func-tion of the beam parameter w0.

Fig. 9. Strength of the longitudinal (Hz ) and radial (H ρ) magneticfields of an APB for two different beam parameters w0 at λ � 523 nm,evaluated using accurate PWS calculations. The strength of the azimu-thal electric field (Eφ) normalized to the wave impedance is also plot-ted for comparison.

Research Article Vol. 33, No. 11 / November 2016 / Journal of the Optical Society of America B 2271

focus (DOF or axial FWHM) of the longitudinal magnetic fieldintensity profile for a converging APB is also shown in Fig. 11as a function of w0, using accurate PWS calculations. For thesake of comparison, we also plot the DOF of the electric fieldintensity profile for a traditional circularly polarized FGB inFig. 11. As w0 increases, the Rayleigh range zR increases as�w0�2 and as a result, the beam waist w varies less with respectto z [see Eq. (2), where w is written as a function of z and zR).Therefore, the DOF is much longer for larger w0, which alsomeans the field features are less tight.

5. SPATIAL MAGNETIC RESOLUTION BELOWTHE DIFFRACTION LIMIT

So far, we have demonstrated that fields in the focal plane of aconverging APB with w0 ≥ 0.5λ are constructed only from thepropagating spectrum and therefore they are diffraction limited(see also Appendix B). We have shown using PWS calculationsin Fig. 8 that the transverse FWHM of the longitudinal mag-netic field intensity and the CFWHM of the total electric fieldintensity for such a converging APB are limited by diffractionto 0.56λ and 0.43λ, respectively. In addition, we have alsoshown that the total magnetic field intensity is less collimatedthan the longitudinal magnetic field due to the presence of thestrong annular-shaped transverse magnetic field. In this section,we aim at enhancing the longitudinal magnetic field of an APBand boosting its spatial magnetic resolution below the dif-fraction limit. To overcome the diffraction barrier, evanescent

waves should be excited. One popular approach to generateevanescent waves, required for achieving spatial resolutions be-low the diffraction limit in microscopy, is to use a subwave-length scatterer [48].

Here, we show that a super-tight magnetic-dominant spot isachieved using a subwavelength-size dense dielectric Mie scat-terer (here, silicon nanosphere) having a “magnetic” Mie reso-nance as a magnetic nanoprobe [41,42]. We adopt an initialparaxial electric field distribution for the APB at a referenceplane z � zr (here, zr � −3.5λ) away from the minimum-waist plane based on Eq. (1), as shown in Fig. 12. Recall thatbased on Appendix B, Figs. 6 and 8 and their correspondingdiscussions in the paper, the APBs with beam parameters largerthan w0 � 0.9λ can be, by a good approximation, representedby Eq. (1). Therefore, we adopt w0 � 0.9λ for the illuminatingAPB to have a tight magnetic field spot. In the previous section,the propagation of such an APB was modeled using the PWSand its accurate minimum-waist plane position and field dis-tributions are calculated. Here, we import the APB’s paraxialtransverse electric field distribution [given by Eq. (1)] intothe finite integration technique in the time-domain solverimplemented in CST Microwave Studio as a boundary fieldsource. As a consequence of the Schelkunoff equivalence prin-ciple (PEC equivalent) implemented in CST MicrowaveStudio, the APB propagates toward the �z direction inFig. 12. The coefficient V in Eq. (2) is set to 0.89 V such thatthe total power of the incident APB given in Eq. (13) is 1 mW.In the full-wave simulations, we assume a free-space wavelengthof λ � 523 nm. The magnetic field map of the incident APB(without any scatterer yet), calculated by the time-domainsolver implemented in CSTMicrowave Studio in the y–z longi-tudinal plane, is shown in Fig. 13(a). We observe fromFig. 13(a) that the APB’s minimum-waist plane occurs atzf ≈ −0.55λ, which is also obtained by the PWS calculationsaccording to Fig. 7 (the paraxial approximation instead wouldestimate a focus at z � 0). Next, a subwavelength-size siliconnanosphere scatterer is placed at the APB’s actual minimum-waist plane (z � zf � −0.55λ), which is assumed to be invacuum. The silicon nanosphere has a relative permittivityequal to εr � 17.1� i0.084 and radius of r � 62 nm suchthat its magnetic Mie polarizability magnitude peaks at λ �523 nm [27].

The total magnetic field’s magnitude in the presence ofthe nanosphere (superposition of incident and scattered fields)

Fig. 11. DOF of the longitudinal magnetic field intensity profilefor a converging APB as a function of the beam parameter w0, evalu-ated using PWS. For comparison, the depth of focus of the electricfield intensity profile of a FGB is also plotted.

Fig. 10. Raytracing model of an APB focusing through a lens.Magnetic field vectors are denoted by blue arrows. Spectral compo-nents with large transverse wavenumbers provide strong longitudinalmagnetic fields.

Fig. 12. Schematic of a converging APB with w0 � 0.9λ illuminat-ing a subwavelength-size silicon nanosphere (as a magnetic nanoprobe)placed at the actual minimum-waist plane of the beam: r � 62 nm,zf � −0.55λ, and λ � 523 nm.

2272 Vol. 33, No. 11 / November 2016 / Journal of the Optical Society of America B Research Article

locally normalized to the magnitude of the incident magneticfield is also shown in the y–z longitudinal plane in Fig. 13(b),where we observe large magnetic field enhancement at the scat-terer cross section and in the vicinity of the scatterer. Note thatthe enhanced magnetic field is strongly localized close to thescatterer with relatively very low side-lobe levels resulting ina very high spatial magnetic resolution. Moreover, starting fromthe surface of the scatterer, the tight magnetic spot extends intothe surroundings and drops rapidly away from the nanosphere’ssurface, revealing the presence of evanescent spectral fields inthe near-field close to the scatterer. The normalized magneticand electric field intensities without the presence of the scat-terer (only the incident APB) at the APB’s minimum-waistplane (at z � zf ) and with the scatterer at two different x-ytransverse planes (at z � zf � r and z � zf � 2r) slightlyaway from the scatterer are also shown in Figs. 14(a) and 14(b).

We observe from Fig. 14(a) that a high-resolution magneticfield is obtained with a total magnetic near-field intensity withFWHM of 0.108λ (0.23λ) and a side-lobe peak of 0.018(0.29), relative to the maximum, at the transverse plane tangentto the sphere at z � zf � r (at the transverse plane one radiusaway from the sphere edge at z � zf � 2r). Note that the in-crease of the relative side-lobe level with the distance z from thenanosphere surface is attributed to the increase of the annular-shaped radial magnetic field strength relative to the longitudinalmagnetic field one (similar trend as in Fig. 9). Figure 14(c) alsoshows the FWHMs of the total and the longitudinal magneticnear-field intensities on x–y transverse planes with z > zf . Thelongitudinal magnetic field remains well collimated with noside lobes, whereas the total magnetic field starts to have sig-nificant side lobes as the distance z from the nanosphere surfaceincreases because of the growth of the radial magnetic fieldcomponent. Therefore, the FWHM of the total magnetic fieldintensity is not reported in Fig. 14(c) for that z range (i.e.,z − zf > 2.5r) where the FWHM would not be a measureof resolution anymore.

The total magnetic field spot areas reported here (i.e.,0.009λ2 and 0.04λ2) are much smaller than the ultimate spotarea obtained for the longitudinal magnetic field of a tightlyfocused APB with w0 � 0.5λ without the nanosphere, whichis 0.25λ2. The enhancement of the total magnetic field withrespect to that of the incident APB at two different transverseplanes is also plotted in Fig. 15(a), where we observe a signifi-cant enhancement of the total magnetic field close to the nano-sphere. The longitudinal magnetic field of the incident APBinduces a magnetic dipole moment in the nanosphere polarizedalong the z direction, which in turn boosts the total magneticfield thanks to the dipolar magnetic near-fields. The square ofthe near-field admittance, defined as the total magnetic fieldintensity divided by the total electric field intensity, normalizedto that of a plane wave (1∕η2) is also plotted in Fig. 15(b),which clearly shows a very high contrast ratio between magneticand electric fields, especially around the beam axis. On the beamaxis (ρ � 0) where the electric field has a null, the magnetic toelectric field contrast ratio goes to infinity (not shown here).

The utilization of a silicon nanosphere as a magnetic nano-probe excited by a converging APB, which provides a tight

Fig. 14. Normalized total (a) magnetic and (b) electric near-fieldintensities (each case is normalized to its own maximum) without(black solid curves) and with (blue dashed and red dotted curves)the presence of the silicon nanosphere centered at z � zf , evaluatedat different x-y transverse planes. (c) The FWHM of the total and thelongitudinal magnetic near-field intensity patterns at different trans-verse planes with z > zf .

Fig. 15. (a) Total magnetic field (summation of incident and scat-tered fields) of the scatter system locally normalized to that without thenanosphere at z � zf , evaluated at different transverse planes awayfrom the scatterer. (b) Ratio of the total magnetic field intensity tothe total electric field intensity of the scatter system normalized to thatof plane wave (this defines the local near-field admittance normalizedto that of the plane wave).

Fig. 13. Full-wave simulation results for the magnitude of (a) theincident magnetic field and (b) the total magnetic field (summation ofincident and scattered field from the nanosphere) locally normalized tothe incident magnetic field.

Research Article Vol. 33, No. 11 / November 2016 / Journal of the Optical Society of America B 2273

magnetic-dominant region with enhanced magnetic and neg-ligible electric near-fields, is especially crucial for the explicitexcitation of photoinduced magnetic dipolar transitions insample matter, with nanoscale precision. Magnetic dipolar tran-sitions are, in general, weak when excited at optical frequenciesand overshadowed by their stronger electric counterparts. Notethat when the nanosphere is utilized as a magnetic nanoprobethat boosts the magnetic response of a sample matter in its closevicinity, the beam axis and the nanosphere should be perfectlyaligned in order to boost selectively the magnetic response ofthe sample matter. This would be advantageous in future scan-ning probe microscopy based on photoinduced magnetismas the extreme sensitivity to alignment would result in high-resolution mapping. In addition, in practical applications,the nanosphere is either deposited on top of a substrate orlocated inside a host. Most importantly, the presence of the di-electric/substrate would (i) shift the focal plane of the converg-ing APB, which could be compensated in a design procedurewhen known, and (ii) change the transverse FWHM of thelongitudinal magnetic field and the CFWHM of the total elec-tric field inside the dielectric medium. Note that the electricfield [which is purely transverse and given in Eq. (3)] and alsothe magnetic field of the APB should be continuous across thedielectric surface according to the field boundary conditions.However, since the effective wavelength inside the dielectricis shorter than that in the vacuum, the field’s features of theilluminating APB would be tighter than those belonging to thesame APB propagating inside unbounded vacuum. Nevertheless,the magnetic-dominant region would be preserved across the di-electric surface. The effect of the dielectric interface on electricfield evolution of tightly focused azimuthally and radially polar-ized beams has been examined in [10,49]; however, its effect onthe magnetic field evolution of the tightly focused APBs shouldbe precisely examined in a future work.

Although in this paper a silicon nanosphere is proposed asan illustrative example of magnetic nanoprobe, other kinds ofthe magnetic nanoprobes (such as circular clusters made ofplasmonic nanoparticles of different geometries [41]) may beadvantageous in terms of increasing the magnetic near-fieldenhancement level, increasing the magnetic-dominant regionsize, or facilitating experimental setups.

6. CONCLUSION

We have characterized the focusing of an azimuthally E-polarizedvector beam (APB) through a lens with special attention on itsmagnetic-dominated region. When focusing the APB, the longi-tudinal magnetic field strength grows relatively more than theazimuthal electric field strength, leading to a region of a boostedlongitudinal magnetic field. We have also elaborated on self-standing converging APBs using PWS calculations and shownthat the longitudinal magnetic field intensity spot with FWHMof 0.56λ and annular-shaped electric field intensity spot withcomplementary FWHM of 0.43λ can be achieved using a con-verging APB with w0 � 0.5λ. However, the resolution of thetotal magnetic field intensity at a diffraction-limited APB focusis limited by the presence of the radial magnetic field in anannular-shaped region around the beam axis with comparablemagnitude to the longitudinal one. In order to enhance the

longitudinal magnetic field and obtain a very high total magneticfield resolution, we have proposed to utilize a magnetically polar-izable (at optical frequency) particle leading to sharp magneticnear-field features. Full-wave simulation results reported heredemonstrate that by placing a subwavelength-size dense Miescatterer (here, a silicon nanosphere) at the minimum-waistplane of a converging self-standing APB, one achieves anextremely high-resolution magnetic-dominant region with amagnetic field enhancement of about 2.3 (with respect to theincident magnetic field) and a magnetic field spot area of0.04λ2 at a transverse plane 0.12λ away from the scatterer sur-face. Such a super-tight magnetic-dominant region, with en-hanced magnetic and negligible electric near-fields, is essentialfor unambiguous excitation of magnetic dipolar transitions inmaterials. This may be beneficial by adding an extra feature,based on magnetic near-field signature, to future magnetism-based scanning probe microscopy and spectroscopy systems.

APPENDIX A: FIELD AT THE FOCAL PLANEOF A LENS UPON APB ILLUMINATION

Let us assume that an incident paraxial APB as in Eqs. (1) and(2) with beam parameter w0;i illuminates an infinitely thin con-verging lens (Fig. 4). We assume the lens to be positioned atz � 0 in the transverse plane where the incident APB has itsminimum CFWHM. In other words, the lens is located at theincident beam’s paraxial minimum-waist plane. Accordingly,the following conclusions would be still approximately validif the incident beam’s minimum-waist plane occurs at jzj ≪ zR ,leading to ζ ≈ 1 and w ≈ w0 in Eq. (2). The electric field at thelens’s paraxial focal plane z � f , on the right side of the lens inFig. 4 is subsequently calculated using the Fresnel diffractionintegral that in cylindrical coordinate system is written as [45]

Ef �ρ;ϕ; z � f � � ki2πf

eikρ2

2f eikfZ

∞

0

Z2π

0

P�ρ 0�×

×

0@Ei�ρ 0;ϕ 0; z 0 � 0�eiΦ�ρ 0�e

�ikρ 02

2f

�1A

× e−ikf ρρ

0 cos�ϕ−ϕ 0�ρ 0dρ 0dϕ 0; (A1)

where f is the lens paraxial focal distance, Ei�ρ 0;ϕ 0; z 0 � 0� isthe incident APB’s electric field vector at the lens plane [givenby Eq. (1) with z 0 � 0], P�ρ 0� is the pupil function to accountfor the physical extent of the lens, andΦ�ρ 0� is the lens-inducedspherical phase given in Eq. (14) required to focus the beam.Under paraxial approximation, the phase term Φ�ρ 0� inEq. (14) is approximated as

Φ�ρ 0� ≈ −kρ 02

2f: (A2)

Substituting Eq. (A2) into Eq. (A1), the Fresnel integral forthe electric field at the lens’s paraxial focal plane can besubsequently approximated as

2274 Vol. 33, No. 11 / November 2016 / Journal of the Optical Society of America B Research Article

Ef �ρ;ϕ; z � f � ≈ ki2πf

eikρ2

2f eikf

×Z

∞

0

Z2π

0

P�ρ 0�Ei�ρ 0;ϕ 0; z 0 � 0�e−i kf ρρ 0 cos�ϕ−ϕ 0�ρ 0dρ 0dϕ 0:

(A3)

For simplicity, let us first assume that the physical diameter ofthe lens (i.e., 2a) is sufficiently larger than the beam waist of theincident beam, implying that almost all the incident beampower illuminates the lens. Under such assumption, the pupilfunction in Eq. (A3) is set to one. The incident APB’s electricfield Ei�ρ 0;ϕ 0; z 0 � 0� in Eq. (A3) is a superposition of fourlinearly polarized LG beams, two x-polarized LG beams with�l ; p� � ��1; 0� and two y-polarized LG beams with �l ; p� ���1; 0� [see Eq. (1)], where l and p are the azimuthal and radialLG beam’s mode numbers, respectively. Therefore, for the sakeof simplicity, we first show the steps for a general linearly po-larized LG beam with mode number �l ; p� and beam parameterw0;i as the incident beam. An analogous treatment is readilyapplied to all four linearly polarized LG beams that formthe APB in Eq. (1). For an incident x-polarized LG beam,the electric field at the lens plane (z � 0) is given asEi�ρ 0;ϕ 0; z 0 � 0� � ul;p�ρ 0;ϕ 0; z 0 � 0�x. Here, we use thefollowing integral identities [47]:Z

2π

0

eilϕ 0e�−ikf ρρ

0 cos�ϕ−ϕ 0��dϕ 0 � 2πil eilϕJ l

�kfρρ 0�;

Z∞

0

ρ 0�jl j�1

2

�e−βρ 02Ljl jp �αρ 02�J l

�kρ 0ρf

��kρ 0ρf

�12

dρ 0

� �sgn�l�jl j2−�jl j�1�β−�jl j�p�1��β − α�p�kρf

��jl j�12

�

× e�−k

2ρ2

4βf 2

�Ljl jp

�αk2ρ2

4βf 2�α − β�

�; (A4)

where J l �·� is the Bessel function of the first kind and order of l ,sgn�l� denotes the sign of l , and Ljl jp �·� is the associated Laguerrepolynomial [1,12]. Accordingly, the x-polarized LG beam’s fieldat the paraxial focal plane of the lens is then written as

Ef �ρ;ϕ; z � f � ≈ V �sgn�l�jl j�−1�pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2p!π�p� jl j�!

sei

kρ22f

× eikfi�l−1�

w0;f

�ρffiffiffi2

p

w0;f

�jl je−�

ρw0;f

�2

Ljl jp

��ρffiffiffi2

p

w0;f

�2�eilϕx; (A5)

where

w0;f � λ

π

fw0;i

: (A6)

For practical cases (as for the lens physical parametersprovided in this paper), almost all the focused beam poweron the lens focal plane is confined in a circular region whosearea is much smaller than πf λ, i.e., the focused area is withina radial distance ρ ≪

ffiffiffiffiffiffiλf

p. Therefore, the phase factor

exp�ikρ2∕�2f � in Eq. (A5) can be neglected and Eq. (A5)would clearly represent a paraxial LG beam at its minimum-waist plane with beam parameter of w0;f . In other words, underparaxial assumption, focusing an incident LG beam through

a simple lens placed at its minimum-waist plane results in an-other LG beam whose minimum-waist plane coincides with thelens’s focal plane (z � f ) and its beam parameter relates to thefocal distance of the lens and the incident beam parameterthrough the Eq. (A6). Note that, in principle and accordingto Eq. (A6), if the radial spread of the incident LG beamdetermined through w0;i is in comparable length to the focusdistance f , then the beam parameter of the converged beamw0;f would be subwavelength. However, as shown in Section 4,physical limitations accounted by using the PWS (see [12] fordetails on the PWS calculations) shows that there is a limit inthe minimum achievable w0;f .

The electric field at the lens paraxial focal plane due to anincident APB is subsequently obtained by the superposition ofthe focal plane fields of its four constitutive linearly polarizedLG beam terms leading to

Ef � Vffiffiffiπ

p 2ρ

w20;f

e−�

ρw0;f

�2

eikf eikρ2

2f φ: (A7)

In order to take into account for the physical extent of thelens, we also consider the following pupil function:

P�ρ 0� ��1 ρ 0 ≤ a;0 ρ 0 > a;

(A8)

where a is the lens radius. The pupil function in Eq. (A8) isexpanded into a summation of Gaussian functions that comein handy in taking the Fresnel integral in Eq. (A3) analytically.Such a pupil function is approximated with a finite summationof basis Gaussian functions as [50]

P�ρ 0� ≈XNn�1

Ane−Bna2ρ 02; (A9)

where complex coefficients An and Bn are, respectively, expan-sion and Gaussian coefficients. It is demonstrated in [50] thatfor N � 10, the pupil function in Eq. (A8) is well representedby Eq. (A9) with proper coefficients given in [50]. By substi-tuting Eq. (A9) into Eq. (A3), the focusing field at the focalplane z � f of the finite-size lens upon illumination by an in-cident x-polarized LG beam with �l ; p� is calculated as

Ef ≈ V �sgn�l�jl jffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2p!π�p� jl j�!

sei

kρ2

2f eikfil−1

w0;f

�ρffiffiffi2

p

w0;f

�jl jeilϕ

×XNn�1

An�1� βn�−�jl j�p�1��βn − 1�pe−�ρ∕w0;f �2

1�βn Ljl jp

��ρ ffiffi2pw0;f

�21 − β2n

�x;

(A10)

where βn � Bnw20∕a2, w0;f is given in Eq. (A6), and the

summation over the Gaussian expansion index n appears inthe focal field distribution term. In this way, the paraxialapproximation of the focusing field at the lens focal planedue to an incident x-polarized LG beam is convenientlyexpressed in series terms of Eq. (A10) for the case of a pupilfunction of finite extent. The electric field at the lens paraxialfocal plane due to an incident APB illumination is then calcu-lated by the superposition of the four constitutive linearlypolarized LG beam terms.

Research Article Vol. 33, No. 11 / November 2016 / Journal of the Optical Society of America B 2275

APPENDIX B: SPECTRAL INTERPRETATIONOF THE BEAM PROPAGATION IN THENONPARAXIAL REGIME

The electric field distribution for APB given in Eq. (1) repre-sents a self-standing beam in the paraxial regime. Therefore, itis important to address limitations of these paraxial expressionsin the cases of beams with very tight spatial extents (small w0).With this goal in mind, we report in Fig. 16 the normalizedmagnitude of the plane-wave spectrum for APBs, i.e., the2D Fourier transform of the transverse field of the APB as

E�kx; ky; z� �Z �∞

−∞

Z �∞

−∞E�x; y; z�e−ikxx−ikyyd xdy (B1)

(see [12] for details on the numerical calculation of the inte-gral). In Fig. 16, we show the wavenumber spectrum ofthree APBs with different beam parameters: (a) w0 � 3λ,(b) w0 � 0.9λ, and (c) w0 � 0.5λ. It is observed fromFig. 16 that the plane-wave spectrum of the tighter beam (beamwith smaller w0) covers a wider region in the kx − ky plane,where k0 � 2π∕λ is the free-space wavenumber. Moreover,the field spectral distribution for all three beams is well con-fined in the propagating wave spectrum with k2x � k2y < k20.Hence, they are mainly constructed by propagating spectralcomponents only. They propagate along the z axis withexp�ikzz�, where kz is real and evaluated as

kz �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik20 − �k2x � k2y �

q: (B2)

All spectral magnitude distributions in Fig. 16 are representa-tive at any z plane as these field spectral distributions propagatewith no magnitude variation (implied by the propagator withmagnitude j exp�ikzz�j � 1).

Let us now consider these three field distributions and lookat the paraxial wave approximation. The paraxial wave equationis valid under the assumption that most of the field spectrum isconfined to a region with k2x � k2y ≪ k20. Under this condition,the accurate PWS evaluation can be approximated with the par-axial field expression of a propagating beam as in Eqs. (1) and(2) [46]. Indeed, the required condition for deriving the para-xial field expressions using PWS calculations is to approximateEq. (B2) as

kz ≈ k0 − �k2x � k2y �∕�2k0�: (B3)

It is observed from Fig. 16 that for w0 � 0.9λ and w0 �0.5λ cases, the spectral distributions cannot be fully confinedto a region with k2x � k2y ≪ k20 in contrast to the case withw0 � 3λ. Therefore, the prediction of the paraxial beam propa-gation is expected to deviate from the actual propagation of thebeam, much more for w0 � 0.5λ than for w0 � 0.9λ andmuch more for w0 � 0.9λ than for w0 � 3λ. The spectra withw0 � 0.5λ and w0 � 0.9λ generate tight field spots, but the zlocation of the tight spots cannot be accurately predicted by theparaxial field equations.

In particular, when considering a converging beam withw0 � 0.5λ or 0.9λ, we can expect that the actual tight spotlocation (minimum-waist plane) will be formed closer to thereference plane than the one predicted by the paraxial expres-sions. This is due to the fact that the field of the APB withw0 � 0.5λ or 0.9λ constitutes plane-wave spectral componentswith large transverse wavenumbers that are not modeled accu-rately in the paraxial field expressions and propagate at largerincidence angles with respect to the beam axis (the z axis) andthus a tight spot forms closer than the one predicted by paraxialfield expressions. The time-average spectral energy of the APBper unit length along the z direction is calculated as

W � 1

4ε0jEj2 �

1

4μ0jHj2: (B4)

We define here the figure of APB’s spectral energy as the ratio ofthe APB’s spectral energy per unit length in the propagatingspectrum to its total spectral energy per unit length:

FW �RR

k2x�k2y <k20W dkxdkyR�∞

−∞R�∞−∞ W dkxdky

: (B5)

Figure 17 shows the figure of APB’s spectral energy as a func-tion of the beam parameter. It is observed that for w0 ≥ 0.5λ,more than 95% of the APB’s spectral energy is confined in thepropagating spectrum.

Funding. W. M. Keck Foundation (USA).

Acknowledgment. The authors would like to thankComputer Simulation Technology (CST) of America, Inc.for providing CST Microwave Studio, which was instrumentalin this work.

Fig. 16. Normalized magnitude of the transverse field spectrum Efor APBs with (a) w0 � 3λ, (b) w0 � 0.9λ, and (c) w0 � 0.5λ. Notethat these APBs are made mainly by the propagating spectrum (suchthat k2x � k2y < k20), and therefore the spectral magnitude profiles arebasically similar at any transverse plane (here, we show only the propa-gating spectrum region).

Fig. 17. Ratio of the APB’s spectral energy per unit length (in the zdirection) confined in the propagating spectrum to its total spectralenergy per unit length (so-called figure of APB’s spectral energy)defined in Eq. (B5) as a function of the beam parameter w0.

2276 Vol. 33, No. 11 / November 2016 / Journal of the Optical Society of America B Research Article

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